A thesis submitted to the Board of Studies in Physical ...

Investigations on discrete Wigner functions of multi-qubitsystems

By

K. Srinivasan

(Enrolment No. PHYS 02 2012 04 005 )

Indira Gandhi Centre for Atomic Research, Kalpakkam, India.

A thesis submitted to the

Board of Studies in Physical Sciences

in partial fulfillment of requirements

for the Degree of

DOCTOR OF PHILOSOPHY

of

HOMI BHABHA NATIONAL INSTITUTE

August, 2017

Chapter 0

ii

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I dedicate this thesis to

Mom, Dad

and

Friends

v

Acknowledgments

My Sincerest and heartfelt gratitude to my guide Prof. G. Raghavan for his inspira-

tional guidance both in academic as well as personal life, throughout these long five

years. I also would like to extend my gratitude to him for teaching me from basics to

careful reading of the papers and thesis.

I would like to thank my mother for showing unconditional care, love and support. I

wish my father were now with me as I am sure he would have been proud of me. I owe

much of what I’m today to the care and love with which he brought me up and for

educating me. I also thank my sisters Radhika and Sudha for their love and affection.

My sincere thanks to my uncle R. Venkatesan for his care and valuable guidance

during hard times. I would like to thank my friends U. Karthikeyan, E. Barath,

K. Sundaravadivelu, S. Sathish, Velmurugan, Jayaprakash, Magesh, M. Karthick,

Thirumalai and all my childhood friends, school mates and college mates for their

unexpected support and innumerable help throughout my life.

I would like to thank my physics teachers Vengatesan and Hubert Dhanasundaram for

their excellent teaching, which motivated me to pursue a degree in Physics. I would

also like to thank my physics Professors Arunchunai Annadurai and A. Santhanam

and Prof. S.V.M. Sathyanarayana for their amazing lectures, discussions and for

being an inspiration for me to carry out research in theoretical physics. I express

my heartfelt thanks to Prof. S. Kanmani for teaching fundamental concepts in linear

algebra and Lie groups, and Prof. S. Sivakumar for his excellent lectures quantum

mechanics and quantum optics. I extend my acknowledgment to Dr. B. K. Panigrahi,

Dr. Sharat Chandra, V. Sridhar and Dr. V. Sivasubramaniam and Dr. V. Sridharan

for their encouragement and support.

I would like to thank Dr. D. Karthickeyan, Dr. P. Anees, Manan Dholakia for their

support, motivation and entertainment throughout these five years. I extend my

vi

thanks to my section members B. Radhakrishna, Dr. Gururaj, Nilakantha Meher and

Dhilipan for their support and endless discussions. I also would like to thank all my

Kalpakkam friends, especially, K.G Ragavendra, L.K. Preethi, Balakrishnan, Mahen-

dran, C. Lakshmanan, Vairavel, T.R. Devidas, Thangam, Nair Radhikesh Ravindran,

Sivakumar, Barath, Surendar for their care and friendship.

I would like to acknowledge DAE for the award of research fellowships and providing

me an excellent accommodation during my course of study. I extend my gratitude to

Dr. A.K. Bhadhuri, Director IGCAR and Dr. G. Amarendra, Director, MSG for their

support. I also like to thank Dr. M. Saibaba for his support and for the provision of

family accommodation, and all his involvement and help towards research scholars. I

would also like to thank Dr. N.V. Chandrasekar, Dean, Physical Sciences for timely

help and support. I would also like to thank the office staff members of MSG and

RMG for their help.

vii

Contents

Page

1 Introduction 1

1.1 Phase space formalism of quantum mechanics-A brief history . . . . . . 1

1.2 Discrete phase space representations for QIP . . . . . . . . . . . . . . . 3

1.3 Discrete Wigner functions and the Wootters’ construction . . . . . . . 4

1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quantum mechanics in a nutshell 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Composite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Linear operators and observables . . . . . . . . . . . . . . . . . . . . . 9

2.5 Trace of an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Mixed states: Density Operators and density matrices . . . . . . . . . . 10

2.7 Partial trace operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.8 Bloch sphere representation . . . . . . . . . . . . . . . . . . . . . . . . 13

2.9 Quantum process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.9.1 Unitary Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.10 Quantum measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.10.1 Projective measurements . . . . . . . . . . . . . . . . . . . . . . 17

2.10.2 POVM measurements . . . . . . . . . . . . . . . . . . . . . . . 18

2.11 Anti-unitary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.11.1 Anti-linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.11.2 Anti-unitary transformation . . . . . . . . . . . . . . . . . . . . 18

2.12 Local operation and classical communication (LOCC) . . . . . . . . . . 19

2.13 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.14 Entanglement detection and quantification . . . . . . . . . . . . . . . . 22

2.14.1 The PPT criterion . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.14.2 The von Neumann entropy . . . . . . . . . . . . . . . . . . . . . 23

2.14.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.14.4 Entanglement of formation . . . . . . . . . . . . . . . . . . . . . 24

2.14.5 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.14.6 Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

viii

3 Introduction to quasi-probability distribution functions 27

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Introduction to Wigner Functions . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Average of an Operator and the Weyl Transform . . . . . . . . 30

3.2.2 Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.3 Wigner functions of the ground and first excited states of aharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Review of Discrete Wigner Functions . . . . . . . . . . . . . . . . . . . 36

3.3.1 Brief survey of DWF constructions . . . . . . . . . . . . . . . . 36

3.3.2 Applications of DWFs . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Gibbons et al., construction over a finite field . . . . . . . . . . . . . . 40

3.4.1 Galois fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.2 Irreducible polynomial . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.3 Trace of the field element . . . . . . . . . . . . . . . . . . . . . 41

3.5 Mutually unbiased bases . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Discrete phase space for prime dimensions . . . . . . . . . . . . . . . . 43

3.6.1 Discrete Wigner Functions . . . . . . . . . . . . . . . . . . . . . 47

3.6.2 Properties of phase space point operators . . . . . . . . . . . . . 49

3.6.3 Properties of Discrete Wigner Functions . . . . . . . . . . . . . 49

3.6.4 Labeling scheme for lines and striations for multiqubit DWFs . 50

3.7 Discrete Wigner function of single qubit systems . . . . . . . . . . . . . 51

3.8 Discrete Wigner function of two qubit systems . . . . . . . . . . . . . . 53

3.8.1 Equivalence classes of quantum nets . . . . . . . . . . . . . . . . 56

3.9 Advantages and limitations of the Gibbons et al. construction . . . . . 58

4 Spin flip of multiqubit states in discrete phase space 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Spin flipped DWF of a multiqubit system . . . . . . . . . . . . . . . . . 63

4.2.1 Derivation of W (∗) for the multiqubit state . . . . . . . . . . . . 63

4.2.2 Proof that F is a Hadamard Matrix and that it is independentof the quantum net . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.3 The spin flipped Wigner function W . . . . . . . . . . . . . . . 66

4.3 Illustration of the spin flip operation for one and two qubit DWFs . . . 67

4.3.1 Illustration for a single qubit system . . . . . . . . . . . . . . . 67

4.3.2 Spin flipped DWF of a two qubit system . . . . . . . . . . . . . 70

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Stokes vector and its relationship to DWF of multi-qubit systems 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Stokes vectors and its relationship to density matrices . . . . . . . . . . 80

5.2.1 For single qubit systems . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 For two qubit systems . . . . . . . . . . . . . . . . . . . . . . . 81

ix

5.2.3 For multiqubit systems . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Stokes vectors and its relationship to DWFs . . . . . . . . . . . . . . . 83

5.3.1 Summary of known results for single qubits . . . . . . . . . . . 83

5.3.2 Generalization to N-qubit systems . . . . . . . . . . . . . . . . . 88

5.3.3 Spin flip operation for n-qubit systems . . . . . . . . . . . . . . 90

5.4 Minkowsky squared norm of an N-Qubit state in terms of the DWF . . 91

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Generalized reduction formula for DWFs of multi-qubit systems 95

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Earlier results on two-qubit DWFs . . . . . . . . . . . . . . . . . . . . 97

6.3 A general reduction formula for two qubit DWFs . . . . . . . . . . . . 99

6.4 Reduction formula for the general multiqubit DWF . . . . . . . . . . . 102

6.4.1 Minkowskian squared norm and the spectrum of the single qubitDWF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.2 Entanglement of two qubit pure states . . . . . . . . . . . . . . 105

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Experimental reconstruction of two-qubit DWF 109

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Spontaneous parametric down-conversion process . . . . . . . . . . . . 110

7.3 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.4 Reconstruction of two-qubit discrete Wigner functions . . . . . . . . . 115

7.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Summary, conclusions and future directions 125

8.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

x

List of Figures

2.1 Bloch sphere representation of qubits. . . . . . . . . . . . . . . . . . . 13

3.1 Wigner function of the ground state of a quantum harmonic oscillator. 34

3.2 Wigner function of the first excited state of a quantum harmonic oscil-lator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Illustration of interference effects due to the superposition of two dis-placed ground states centered at q = +b and q = −b. . . . . . . . . . . 34

3.4 Wigner function of a quantum system, which is a statistical mixture oftwo coherent states centered at q = +b and q = −b. . . . . . . . . . . . 35

3.5 (a) Labeling the points of 2 × 2 phase space by a finite field F2. (b)Labeling the lines of the discrete phase space with pure states. (c)Lines and striations of the 2 × 2 phase space, set of orange points arerays and the set of black points are other line of the striations. . . . . . 53

3.6 (a) Labeling discrete phase space points by elements of the finite fieldF4. (b) Labeling horizontal and vertical lines of the phase space bypure states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7 Lines and striations of 4× 4 discrete phase space. . . . . . . . . . . . . 54

3.8 Figure shows two possible equivalence classes available in a 2×2 phasespace and each equivalence class contains four quantum nets. Withinthe equivalence class, two distinct quantum nets are related to eachother by unitary transformations {σx,σy,σz}. . . . . . . . . . . . . . . 56

5.1 Density matrix, multiqubit Stokes vector and the DWF are valid repre-sentations of the multiqubit systems. Figure shows the possible trans-formation formulas available to connect one representation to the otherand the unavailability of the direct relationship between the DWF andthe Stokes vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 Two cemented BBO crystals are used to generate polarization entan-gled photons, whose optics axes are oriented 90◦ with respect to eachother. Two overlapping cones are produced, one in each crystal. . . . . 112

7.2 Experimental set up to generate a pair of polarization entangled pho-tons using spontaneous parametric downconversion. . . . . . . . . . . . 113

7.3 A point α in the 4× 4 phase space φ can be identified by the set of twopoints α1 and α2 corresponding to φ1 and φ2 respectively. Therefore,every point α in phase space can be written as an ordered set of phasespace points α = (α1, α2). For example, the point α = ((1, 0), (0, 1)) inφ is the point (0, 1) in the square (1, 0). This is shown as a dot in thefigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

xi

7.4 Experimentally reconstructed DWF of bi-photon polarization entan-gled state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5 Real part of the experimentally reconstructed density matrix of bi-photon polarization entangled state. . . . . . . . . . . . . . . . . . . . . 118

7.6 Imaginary part of the experimentally reconstructed density matrix ofbi-photon polarization entangled state. . . . . . . . . . . . . . . . . . . 119

xii

List of Tables

3.1 Labeling scheme for striations . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Addition and multiplication table for F4 . . . . . . . . . . . . . . . . . 54

3.3 The mutually unbiased basis vectors associated with lines of the 4× 4phase space. This table provides a fixed point to generate the rayassociated with each striation and also the unitary operators that leavethe lines of the striation invariant. . . . . . . . . . . . . . . . . . . . . . 56

4.1 The DWF of W ∗ subjected to rigid translation effected by σy ⊗ σy,results in the shifting of the elements of W ∗ by β = (ω, ω) to yield thecorresponding element of W . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 W I , WX , W Y and WZ are DWFs of the 2 × 2 identity matrix andPauli matrices σx, σy, σz respectively. The sum of the all the DWFsis related to the trace of the corresponding operator,

∑α

W iα = Tr(σi),

where i ∈ [I, x, y, z]. Therefore, the sum of DWF elements of the W I

is 2 and the sum is 0 for the DWF of the Pauli operators. . . . . . . . 85

7.1 Realizing different single qubit projection operators using quarter waveplate and polarizers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 Single and coincidence counts (CC) for 36 projective measurements. . . 120

7.3 Coincidence counts associated with the 9 measurements, each measure-ment consisted of 4 projective measurements. For example, coincidencecounts given in the first line is associated with the σz⊗σz measurement. 120

7.4 Reconstructed DWF of the two photon polarization entangled states. . 121

7.5 DWF associated with the spin flipped system. . . . . . . . . . . . . . . 121

7.6 DWF of the sub-systems A and B computed from the reduction formula.121

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Publications Chapter 0

xiv

Chapter 1

Introduction

1.1 Phase space formalism of quantum mechanics-

A brief history

Quantum states are represented by vectors in a Hilbert space or more generally by a

convex set of positive-definite operators of unit trace, called density operators [1, 2].

Quantum states, thus represented, are rather abstruse as compared to classical states

of a non-relativistic system defined to be points in phase space. For a conservative sys-

tem, the dynamics of such states is obtained using the Hamilton’s equations of motion.

At this level of description, the state is ontological in character and its time evolution

describes a deterministic trajectory in phase space. However, when the degrees of

freedom of the classical state are very large or when only incomplete information is

available about the initial conditions, the classical system is best described in terms

of a probability density distribution ρ (q, p) and dynamics is described by the classical

Liouville equation ddtρ = {ρ,H}, where the braces denote the Poisson brackets [3].

Time evolution of a quantum state is dictated by the Schrodinger equation or more

generally by the equation ddtρ = i

~ [ρ,H ]. This equation bears a certain similarity

with the classical Liouville equation save for the appearance of the ih

term and the

classical Lie bracket being replaced by the quantum one - the commutator [4]. This

resemblance not withstanding, ρ is a function of neither the position nor the momen-

tum operator. Further, these operators do not commute with each other and hence

1

Phase space formalism of quantum mechanics-A brief history Chapter 1

the position and momentum eigenvalues cannot be simultaneously measured. Thus,

the quantum description of the state and its evolution lacks the the intuitive appeal

of the classical phase space description. However, the density operator is by no means

a unique description of the quantum state, as it is possible to formulate quantum

mechanics in terms of phase space distributions. Such a formulation is of substantial

interest both in applications and issues related to the foundations of quantum mechan-

ics like the emergence of classicality [5], derivation of semi-classical equations [6] and

the study of stochastic dynamics [7]. In fact, innumerable such distribution functions

can be defined. The Wigner function, Husimi Q function and Sudharshan-Klauder

P -distributions are some well known distribution functions of this type [4, 8–12]. It

must be stated however, that valid probability distribution functions which depend on

both position and momentum do not exist because their associated operators do not

commute with each other [4]. For instance, Wigner functions are called quasiproba-

bility rather than probability distributions because they can take negative values in

some regions of the phase space but are however real-valued and normalized. Classical

states of light like coherent states have positive Wigner functions [13] but, this is not

the case for quantum states of light such as photon added/substracted coherent states

or entangled states [14–17]. In fact, negative values of the Wigner function attest to

the quantum character of the state [18]. As against standard quantum mechanics, the

quantum phase space formalism has the advantage of providing an intuitive connection

with classical mechanics. In fact, starting from the evolution equation for the density

operator, it is possible to derive an equation of motion for the Wigner function that

is a version of a “Quantum Liouville equation”. In the classical limit, this equation

goes over to the classical Liouville equation. The Quantum Liouville equation can

be derived through functions called Moyal functions [4]. Moyal’s own motivation was

the reformulation of quantum mechanics in statistical terms in order to gain a better

insight into its very formalism. The important point is that this new formulation is

independent of any reference to the density operator and the Schrodinger equation [8].

2

Chapter 1: Chapter 1

In terms of modern applications, Wigner functions are extensively used in quantum

optics as they help to visualize oscillatory photon statistics through exquisite quan-

tum interference effects [19]. From an experimental perspective, Wigner functions can

be reconstructed through homodyne measurements and quantum interference effects

are quite nicely brought-out in the visual presentation of the reconstructed state [20–

23]. Next section discusses the phase space representation of the finite dimensional

quantum systems.

1.2 Discrete phase space representations for QIP

Phenomenal advances have been made in recent years in the fields of Quantum In-

formation processing (QIP) and Quantum computation. These areas exploit two

distinctive properties of quantum states viz. superposition and entanglement as novel

resources for realizing tasks which are either not acheivable with classical states or

could be done better with these resources [24–27]. For most problems in quantum

information processing, rather than continuous systems, quantum states in a finite

dimensional Hilbert space are of interest. In fact, the quantum state of a two level

system, called the Qubit, is a fundamental unit of information [28]. The qubit is a

vector in a two dimensional Hilbert space. The state of larger systems called mul-

tiqubit systems are constructed from the tensor product of two dimensional Hilbert

spaces [1, 2]. Given the success of Wigner functions for continuous systems, there was

a natural interest in developing its discrete analogue. These are basically a normalized

set of real numbers distributed over a two dimensional grid of points and are called

Discrete Wigner Functions (DWFs) [29–38]. These representations find applications

in quantum error correction [39, 40], quantum teleportation [41, 42], study of decoher-

ence [43] and in the construction of toy models in support of epistemic interpretations

of the quantum state [44]. Another important result in quantum computation shows

that only pure states with non-negative DWFs can be simultaneous eigenstates of

3

Discrete Wigner functions and the Wootters’ construction Chapter 1

generalized Pauli operators [45]. These states are called stabilizer states and it has

been shown that such states and their unitaries which preserve the non-negativity are

amenable to efficient classical simulation [46, 47].

1.3 Discrete Wigner functions and the Wootters’

construction

Several alternate constructions of the Discrete Wigner Functions (DWFs) have been

presented in the literature [29–38]. The construction given by Wootters and Gibbons

et al. is employed in the the present work [31–34]. In this construction, the discrete

phase space for an N-dimensional system is a N×N grid of points defined over a finite

Galois field and as a consequence, the discrete phase space has several geometrical

similarities to the Euclidean plane. It is possible to define the lines and the set of

parallel lines called striations in phase space. Quantum states are mapped to the

lines in the discrete phase space and the states associated with the lines of different

striations are mutually unbiased with respect to each other. The DWF is defined in

this discrete phase space by N ×N array of real numbers. Since our own interest lies

in systems with dimension N = 2n, this does not impose any restrictions. However,

even within this construction, a given density matrix have several versions of the

DWF associated with it. The DWF representations of the quantum state depend

on the particular way of assigning Mutually Unbiased Basis Sets (MUBS) [32, 48]

to “lines” in the discrete phase-space. These are called quantum nets, with different

assignments leading to different versions of the DWF. DWFs have been experimentally

reconstructed from an ensemble of measurements using MUBS [49, 50] and, the version

of DWF so reconstructed, depends on the chosen assignment. For Hilbert space of

dimension N , there are NN+1 possible quantum nets, that is NN+1 possible definitions

of DWF corresponding to the same density matrix. The wider use of DWFs is inhibited

because of two important limitations: 1.The DWF representation of the quantum state

4


ρ is not unique. 2. Subsystem information cannot be easily obtained as in the case

of density matrices and Stokes vectors. In the present work, Wootters’ construction

of DWF is investigated for multiqubit states to bring out some new features. This

thesis provides some results which are not present in the literature.

1.4 Outline of the thesis

The present thesis is structured into eight chapters of which this introduction is the

first one. A brief summary of the other seven chapters follows. Chapter-II provides

a concise introduction to the mathematical formulations of quantum mechanics and

quantum information processing as are relevant for the presentation of the work.

Chapter-III provides a brief introduction to continuous Wigner functions and touches

upon its properties and its usefulness in quantum optics. For finite dimensional sys-

tems, the discrete version of the Wigner functions has several distinct constructions.

Some of these constructions are briefly reviewed in this chapter. We then provide

the complete details of the Wootters’ and Gibbons et al., construction with sufficient

tools required for the derivation of the central results presented in this thesis. In the

next three chapters, we address several problems associated with DWFs of multi-qubit

systems. Firstly, in chapter-IV, we supply a method for carrying out the anti-unitary

operation of spin-flip on the DWFs of multi-qubit systems. Based on this result, we

provide a formula for quantifying entanglement of a multi-qubit system directly in

terms of the DWF in chapter-V. The relationship between the Stokes vector repre-

sentation and DWF for different choices of the quantum nets is exploited for this

purpose. Another important problem of interest arises with respect to obtaining the

states of the sub-systems of a composite quantum state. For density matrices, this

is easily done through the partial trace operation [51]. Though such a requirement

arises in several important contexts, a procedure analogous to the partial trace is not

available for DWFs of arbitrary dimensions and for arbitrary quantum nets. This

5

Outline of the thesis Chapter 1

problem has hitherto been addressed only for two qubit systems and restricted to

only certain versions of the DWF [52]. As stated earlier, the assignment of MUBS

is not unique and each assignment gives rise to a distinct quantum net [31] and this

problem gets carried over to the issue of obtaining the partial trace. Therefore, the

partial trace formula should work for all possible quantum nets of the global system

as well as those of the subsystems. In chapter-VI, we exhibit a general prescription

for obtaining sub-systems DWFs from an arbitrary multi-qubit DWF. In chapter-VII,

as an experimental illustration of the reconstruction of the DWF, entangled bipartite

photonic states are generated through the spontaneous parametric down conversion

(SPDC) and their DWFs are reconstructed using quantum state tomography. De-

tails of the experimental setup developed as a part of this thesis are presented in this

chapter.

In summary, the present work builds on the DWF construction of Wootters’ et al.

and presents several new results. The basic idea was to take well known results for

density matrices and generate an equivalent one for DWFs. These developments are

consistent with the spirit of treating quasi-probability distribution in phase space as

an independent formalism of the quantum state in its own right. It is hoped that the

results presented in the thesis would provide the fillip for further explorations and

development of the discrete phase-space formalism.

6

Chapter 2

Quantum mechanics in a nutshell

This chapter provides a brief introduction to essentials of quantum mechanics and

quantum information processing as are relevant for the presentation of the work. The

basic theory related to entanglement and entanglement measures are also discussed.

2.1 Introduction

A vector space H defined over a complex field is called a Hilbert space if it is equipped

with an inner product 〈.|.〉 and is complete in the norm 〈ψ|ψ〉 for |ψ〉 ∈ H. Every quan-

tum mechanical system is in general associated with an infinite dimensional Hilbert

space. If the Hilbert space has N linearly independent vectors that constitutes a span-

ning set, the Hilbert space is N dimensional and is denoted by HN . If N is finite, then

the Hilbert Space is finite dimensional else not. Standard quantum theory is founded

on Hilbert Spaces and operators. For most part, quantum information theory and

quantum computation rely on finite dimensional vector spaces and we shall confine

ourselves to the same hereafter.

2.2 Pure States

A pure state of a quantum system can be represented mathematically as a vector

|ψ〉 ∈ HN . If {|ui〉} be a basis for HN , then any |ψ〉 may be written as:

7

Composite systems Chapter 2

|ψ〉 =∑i

ci|ui〉 (2.1)

where ci’s are probability amplitudes and |ci|2 gives the probability of finding the

system in the state |ui〉 and∑i

|ci|2 = 1. Given a basis choice, a pure state can be

represented by a N × 1 column vector with the coefficients ci as entries. Quantum

states which cannot be represented as a vectors inHN are known as mixed states. The

simplest pure state is a vector |ψ〉 = c1|u1〉 + c2|u2〉 ∈ H2 called a qubit which, can

be realized physically through any two state quantum system. Polarization degrees of

freedom of photons, spin-12

particles and two level atomic systems are some possible

implementations of qubits. The basis Bs =

1

0

,

0

1

is known as the stan-

dard/computational basis and a general qubit can be represented as |ψ〉 =

c1

c2

.

Considering polarization qubits, horizontal and vertical polarization states |H〉 and

|V 〉 can be used as a basis to represent an arbitrary pure state as,

|ψ〉 = cos

(θ

2

)|H〉+ eiφsin

(θ

2

)|V 〉 (2.2)

where θ is an angle with respect to the horizontal polarization and φ is an arbitrary

phase factor.

2.3 Composite systems

The Hilbert space of a composite system is a tensor product of the Hilbert spaces

of the component systems. For example, for bi-partite systems made of component

systems with Hilbert spaces Hd1and Hd2 , the Hilbert space of the composite system is

Hd1×d2 = Hd1⊗Hd2 . As an example, consider the situation that we need to specify the

state of more than one system, say polarization state of two photons. Each photon

8


can be represented by a vector in a two dimensional Hilbert space H2. Therefore,

the Hilbert space for the polarization state of the two photons are constructed by

the tensor product “⊗” of the Hilbert spaces of its subsystems, H4 = HA2 ⊗ HB

2 ,

where HA2 and HB

2 are the Hilbert spaces of the sub-system A and B respectively.

The two-photon states are elements of the 4-dimensional Hilbert space. Representing

both photons in the standard basis Bs =

|H〉 =

1

0

, |V 〉 =

0

1

the general

states of the composite systems is,

|ψAB〉 = cHH |H〉 ⊗ |H〉+ cHV |H〉 ⊗ |V 〉+ cV H |V 〉 ⊗ |H〉+ cV V |V 〉 ⊗ |V 〉 (2.3)

|ψAB〉 = cHH |HH〉+ cHV |HV 〉+ cV H |V H〉+ cV V |V V 〉 (2.4)

where |ψAB〉 ∈ H4 and |HH〉 stands for |H〉 ⊗ |H〉. In general, consider the two

particle system and let HA and HB be the Hilbert spaces of the first and second

sub-systems of dimensions dA and dB respectively. Let {|ui〉} and {|vj〉} be the bases

for the sub-systems A and B, then general bipartite pure states can be written as,

|ψAB〉 =∑i,j

cij|ui〉 ⊗ |vj〉 =∑i,j

cij|uivj〉 (2.5)

where{|ui〉 ⊗ |vj〉}is a basis set in HAB. The state vector representation is not the

most general description of the quantum states. More generally, arbitrary quantum

states are represented by the density matrices. These are discussed in the next section.

2.4 Linear operators and observables

A linear operator L is a map between two vector spaces, L : V → W for all u, v ∈ V

and α, β ∈ F ,

L (αu+ βv) = αL (u) + βL (v) (2.6)

9

Trace of an operator Chapter 2

where L (|u〉) ,L (|v〉) ∈ W . An operator O is said to be an observable, if it is

Hermitian(O† = O

)and its eigenvectors forms a basis for the state space.

2.5 Trace of an operator

Given a representation in a basis {|ϕm〉} the trace of an operator A is given by,

Tr (A) =∑m

〈ϕm|A |ϕm〉. Thus, the trace of the operator is the sum over the diagonal

elements of the matrix representing the operator.

2.6 Mixed states: Density Operators and density

matrices

The state vector is not the most general representation of all possible quantum

states. For example, partially polarized light cannot be represented as a vector in

H2 but, requires a more general description. Quantum states are more generally

represented by positive semi-definite operators of unit trace called density operators.

Considering collection of all pure states |ψi〉 ∈ H represented as rank one projectors

{ρi : ρi = |ψi〉〈ψi|}, the state space S(H) is defined to be the convex set,

ρ =∑i

pi|ψi〉〈ψi| (2.7)

ρ =∑i

piρi (2.8)

where pi ≥ 0,∑i

pi = 1. Thus, the general quantum state is a statistical mixture

of pure states, which cannot be represented as a vector in a Hilbert space. We may

note that the decomposition of a density operator is not unique. In summary, density

matrices have the following properties,

1. They are self adjoint: ρ = ρ†.

2. Tr(ρ) = 1.

10


3. Tr(ρ2) ≤ 1, for pure states Tr(ρ2) = 1 and for the mixed states, Tr(ρ2) < 1.

4. ρ ≥ 0 i.e, all its eigenvalues λi ≥ 0.

For a given a basis in HN , ρ is N × N matrix, uniquely determined by N2 − 1 real

numbers. Let A be some observable, the expectation value of this observable A in

the quantum state ρ can be calculated as 〈A〉 = Tr(Aρ).

Consider the qubit |ψ〉 represented in the standard basis as,

|ψ〉 = c1|0〉+ c2|1〉 (2.9)

This state can also be represented as a density matrix by,

ρ = |ψ〉〈ψ| (2.10)

ρ = |c1|2|0〉〈0|+ c1c∗2|0〉〈1|+ c∗1c2|1〉〈0|+ |c2|2|1〉〈1| (2.11)

In the standard basis,

ρ =

|c1|2 c1c∗2

c∗1c2 |c2|2

(2.12)

where the diagonal entries are the populations of the states |0〉 and |1〉, that is the

probability of finding the system in the basis states |0〉 and |1〉. The off-diagonal

entries are the coherence between the basis vectors. Therefore, in general single qubit

systems are represented as the 2× 2 density matrices.

2.7 Partial trace operation

Given the density operator of a composite system, the state of the sub-systems can be

easily obtained through the partial trace operation. To illustrate this procedure for a

bi-partite systems, let S(HA) and S(HB) be the state space for the systems A and B,

11

Partial trace operation Chapter 2

and S(HA ⊗ HB) = S(H) be the state space for the composite system. The partial

trace operation maps an element of S(HA ⊗HB) to either S(HA) or S(HB). So the

partial trace over system-B is the map TrB : S(HA ⊗ HB) → S(HA), similarly,

partial trace over system-A is a map TrA : S(HA ⊗HB) → S(HB).

Thus, if the density operator of the composite system is ρAB, then the reduced density

operator of the sub-system A is given by,

ρA = TrB(ρAB) (2.13)

where TrB refers to partial trace over the sub-system B. In general, if the global

system ρAB is a pure state, its sub-systems are not necessarily a pure state, it may

be a mixed state. As an example consider the polarization state of two photons as,

|ψAB〉 =1√2

(|HH〉+ |V V 〉) (2.14)

then the composite system is ρAB = |ψAB〉〈ψAB|. Sub-systems ρA and ρB are given

by,

ρA = TrB(ρAB) =1

2(|H〉〈H|+ |V 〉〈V |) (2.15)

ρA =I22

(2.16)

and

ρB = TrA(ρAB) =1

2(|H〉〈H|+ |V 〉〈V |) (2.17)

ρB =I22

(2.18)

where I2 is a 2 × 2 identity matrix, that is the sub-systems are maximally mixed

states.

12


2.8 Bloch sphere representation

Figure 2.1 – Bloch sphere representation of qubits.

A Bloch sphere is a geometrical representation of the Qubits as a point on or in the

interior of a three dimensional unit sphere. We know that the general quantum system

in the two dimensional Hilbert space, that is qubit, is represented as 2 × 2 density

matrix. As mentioned earlier, 3 real entries are required to represent the general 2×2

density matrix. But, only 2 real entries are sufficient to represent a general pure single

qubit systems. Arbitrary pure single qubit systems can be represented as,

|ψ〉 = cos

(θ

2

)|0〉+ eiφsin

(θ

2

)|1〉 (2.19)

Here, real entries θ and φ uniquely fixes the pure single qubit systems, takes values

0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These normalized pure states can be represented as

point on the unit three dimensional sphere, known as Bloch sphere. Mixed states

are the points inside the unit sphere. Three entries (r, θ, φ) are required to represent

the general mixed states. The completely mixed state is at the center of the sphere.

As shown in Fig (2.1), in the Bloch sphere representation, orthogonal states takes

diametrically opposite points [51].

13

Quantum process Chapter 2

2.9 Quantum process

In standard quantum mechanics, there are two kinds of process one consider. The

first is the unitary process, which governs the dynamics of the quantum system and

the second one is the non-unitary measurement process.

2.9.1 Unitary Evolution

The time evolution of the closed quantum systems |ψ〉 ∈ H is completely described

by the Schrodinger equation,

i~d|ψ〉dt

= H|ψ〉 (2.20)

where ~ is a Planck constant and H is the Hamiltonian, dynamics of the systems is

completely determined, if the Hamiltonian is known. The solution of the Schrodinger

equation provides a relationship between the state of the system |ψ(t1)〉 at time t1 to

the state of the system |ψ(t2)〉 at later time t2, through the unitary operator,

|ψ(t2)〉 = exp

(−iH(t2 − t1)

~

)|ψ(t1)〉 (2.21)

|ψ(t2)〉 = U |ψ(t1)〉 (2.22)

where U = exp(−iH(t2−t1)

~

)is a unitary operator. Therefore, the evolution of a

quantum systems are completely specified by the unitary operation. A linear operator

U is a unitary operator if it satisfies,

UU † = U †U = I (2.23)

or

U−1 = U † (2.24)

Unitary operatorU preserves the transition probabilities, that is 〈ψ1|ψ2〉 = 〈ψ1|U †U |ψ2〉.

14


For any Hermitian operator H , the operator U = eiH is an unitary operator.

2.10 Quantum measurement

“The measurement problem” has evoked much debate amongst researchers working

on the foundations of quantum mechanics. However sticking to the standard version

of quantum mechanics, for every observable A and the state ρ, the measurement

process assigns a probability measure µA on R. This quantity 0 ≤ µ(E) ≤ 1 yields

the probability that the measurement of the observable A results in E ⊆ R. Three

components are important in every measurement process (1) the system (2) the mea-

suring device and (3) the environment surrounding measuring device. In standard

quantum mechanics, the measuring apparatus is always treated as a classical system.

Measurements are devised to measure physically observable quantities like position,

momentum, energy etc., represented by self-adjoint operators. An operator O is said

to be an observable, if it is a Hermiatian operator and its eigenvectors forms a basis

for the state space. Let {|un〉} be the orthonormal set of eigenvectors and {on} are

the eigenvalues of the operator O, then,

〈un|un′〉 = δnn′ (2.25)

and ∑n

|un′〉〈un| = I (2.26)

A measurement process perturbs the quantum systems, rendering the state of the

system into one of its eigenstates. The only exception to this rule is when the initial

state of the system is already in one of the eigenstates. The expectation values of an

observable O in the state |ψ〉 is given by,

〈O〉 =〈ψ|O|ψ〉〈ψψ〉

(2.27)

15

Quantum measurement Chapter 2

If the state |ψ〉 is normalized, then 〈O〉 = 〈ψ|O|ψ〉. Let {|un〉} be the eigenstates and

{on} are the eigenvalues of the observable O, then the expectation value is given by,

〈O〉 =∑n

on|〈un|ψ〉|2

〈ψ|ψ〉(2.28)

〈O〉 =∑n

onPn (2.29)

where Pn = |〈un|ψ〉|2〈ψ|ψ〉 is the probability to find the system in the eigen state |un〉. For

a general quantum system represented as the density matrix, the expectation value

of an observable O can be calculated as,

〈O〉 = Tr(Oρ) (2.30)

In this section, we provide a brief introduction to the general measurement process in

quantum theory. In general, the collection of measurement operators {Mm}, which

satisfies the completeness relation∑m

M †mMm = I, completely describes the quantum

measurement. The subscript m refers to particular outcome of the measurement.

Let |ψ〉 be the initial state of the system, then after the measurement, the outcome

m occurs with the probability p(m) = 〈ψ|M †mMm|ψ〉. The completeness relation

ensures that the sum of all the probabilities p(m) is one,

∑m

p(m) =∑m

〈ψ|M †mMm|ψ〉 = 1 (2.31)

The post measurement state of the system is identified as,

Mm|ψ〉√〈ψ|M †

mMm|ψ〉(2.32)

16


2.10.1 Projective measurements

Projective measurements are the simplest type of measurements in quantum mechan-

ics. In such measurements, the measurement operators are rank one projectors Pm

and are therefore Hermitian P †m = Pm and idempotent P 2m = Pm. Projectors Pm

belongs to the eigenspace of an observable M with the eigenvalue m corresponds to

the possible outcomes. Using spectral decomposition,

M =∑m

mPm (2.33)

The projectors satisfy, PmPn = δmnPm and∑m

Pm = I. A projective measurement

is called complete if the projectors Pm are all one dimensional. Let |ψ〉 be the initial

state of the system, the probability for the particular outcome m is given by,

p(m) = 〈ψ|Pm|ψ〉 (2.34)

and collapses the state |ψ〉 into,

|ψ′〉 =Pm|ψ〉√〈ψ|Pm|ψ〉

(2.35)

If initial state of the system is represented as the density matrix ρ, then p(m) =

Tr (PmρPm) = Tr (Pmρ) and the post-measurement state ρ′ is given by,

ρ′ =PmρPm

Tr (PmρPm)(2.36)

Projective measurements are very useful in distinguishing the orthogonal states. How-

ever non-orthogonal states cannot be distinguished by the projective measurements,

in such case POVM measurements are very useful.

17

Anti-unitary operations Chapter 2

2.10.2 POVM measurements

POVM measurements are more general measurements, and they include projective

measurements as a special case. The elements of the POVM are any set of positive

operators Em, satisfy∑m

Em = I. For every POVM {Em}, there exists measurement

operators Mm defining a measurement such that Mm =√Em and

∑m

M †mMm =∑

m

Em = I.

2.11 Anti-unitary operations

2.11.1 Anti-linear maps

Let V and W are to two complex vector spaces, map f is said to be anti-linear map,

if,

f(ax+ by) = af(x) + bf(y) (2.37)

for all a, b ∈ C and all x, y ∈ V , where a and b are complex conjugates of a and b.

2.11.2 Anti-unitary transformation

An anti-unitary operation U is a bijective map, U : H1 → H2 such that,

〈Ux,Uy〉 = 〈x, y〉 (2.38)

for all x, y ∈ H1. If H1 = H2, then U is an anti-unitary operator.

2.11.2.1 Properties of anti-unitary operators

1. An anti-unitary operator is an invariance transformation of the complex Hilbert

space, which leaves the absolute value of the scalar product invariant,

|〈Ux,Uy〉| = |〈x, y〉| (2.39)

for all x, y ∈ H.

18


2. For all x, y ∈ H, 〈Ux,Uy〉 = 〈x, y〉 = 〈y, x〉.

3. For a anti-unitary operator U , U 2 is an unitary operator,

〈U 2x,U 2y〉 = 〈Ux,Uy〉 = 〈x, y〉 (2.40)

4. The complex conjugation operation K is an anti-unitary operator.

5. For any unitary operator U and the complex conjugation operation K, the

product UK is an anti-unitary operator. Similarly for any anti-unitary operator

V and the complex conjugation operation K, the product VK is an unitary

operator.

2.12 Local operation and classical communication

(LOCC)

Consider the composite quantum system shared between two parties Alice and Bob,

who are separated in space and each have access to one part of the composite system.

Alice can perform measurement on her system associated with the Hilbert space HA

similarly Bob has access to his system defined in the Hilbert spaceHB. The composite

system Hilbert space is given by H = HA ⊗ HB. A more general class of quantum

operation is when Alice performs operations locally on her system and communicates

the measurement outcomes through a classical channel to Bob. Based on the outcome

of Alice’s measurements, Bob acts independently on his system and communicates his

results to Alice through classical channel. This class of quantum operations is known

as local operations with classical communication (LOCC) [53].

19

Entanglement Chapter 2

2.13 Entanglement

Quantum entanglement is a property of composite quantum systems. Rather than a

general definition, it is convenient to illustrate this property based on pure states of

a two qubit system. In the previous section, we have specified the polarization state

of a two photon systems in the 4-dimensional Hilbert space. A general two photon

polarization state can be written as,

|ψ12〉 = cHH |HH〉+ cHV |HV 〉+ cV H |V H〉+ cV V |V V 〉 (2.41)

Let H12 and H2

2 be the Hilbert spaces of the first and the second photons. The state

of the first and the second photons are completely specified by,

|ψ1〉 = a|H〉+ b|V 〉 (2.42)

and

|ψ2〉 = c|H〉+ d|V 〉 (2.43)

Therefore, from the above equations, the state of two photon system can be written

as,

|ψ12〉 = |ψ1〉 ⊗ |ψ2〉 (2.44)

|ψ12〉 = ac|HH〉+ ad|HV 〉+ bc|V H〉+ bd|V V 〉 (2.45)

But later we show that not all states can be represented as product of state vector of

individual systems. States which can be written in product form |ψ12〉 = |ψ1〉 ⊗ |ψ2〉

are known as the product states or the separable states. These states can be prepared

locally by two independent experimenters using LOCC operations. Let

|ψ12〉 =1√2

(|HH〉+ |V V 〉) (2.46)

20


be the state of the two photon system. This system cannot be written in product

form. Because, from Eq (2.41) and Eq (2.45), it is clear that,

ac =1√2

=⇒ a 6= 0 & c 6= 0

ad = 0 =⇒ a = 0 or d = 0

bc = 0 =⇒ b = 0 or c = 0

bd =1√2

=⇒ b 6= 0 & d 6= 0

this is a contradiction. Therefore, in the case of composite systems, there are some

states which cannot be represented as the product of the individual subsystems,

|ψ12〉 6= |ψ1〉 ⊗ |ψ2〉. Such states are known as non-product or entangled states and

they cannot be prepared using LOCC . If the state |ψ12〉 is maximally entangled state

of the two qubit systems then its sub-systems are maximally mixed states. Such states

have the special property that the measurement on the subsystems yields |H〉 and |V 〉

with equal probability 12, that is, there is no knowledge about the sub-system. In their

famous paper, EPR had pointed out that these states have the very strange property

of non-local correlations, which cannot be realized in any classical system. For the

state defined in Eq (2.46), measurement along {|H〉, |V 〉} basis for both the photons

collapses the state |ψ12〉 into either |HH〉 or |V V 〉. If the first system is found to be in

|H〉 then immediately after measurement second system collapses to |H〉 irrespective

of their distance of seperation. These states have been named as Bell state in honor

of J.S.Bell. There are four important maximally entangled states in two qubits which

forms a basis for the two qubit systems, called Bell states,

|φ±〉 =1√2

(|HH〉 ± |V V 〉) (2.47)

21

Entanglement detection and quantification Chapter 2

and

|ψ±〉 =1√2

(|HV 〉 ± |V H〉) (2.48)

For mixed states, a bipartite state ρ represented in the Hilbert space H = HA⊗HB is

known as classically correlated or separable state if and only if it is statistical mixture

of product states [54], that is,

ρ =∑i

piρ(i)A ⊗ ρ

(i)B (2.49)

where pi ≥ 0 and∑i

pi = 1. The states ρ(i)A and ρ

(i)B are an elements of the state space

S(HA) and S(HB). These separable states can be prepared just by LOCC operation.

The states which cannot be written in the separable form given Eq (2.49) are known

as entangled states.

2.14 Entanglement detection and quantification

From the last section, it is clear that multipartite quantum systems can have the

special property of quantum entanglement. The detection and the quantification of

entanglement is an important problem in quantum information theory.

2.14.1 The PPT criterion

Partial transposition is an entanglement witness operator [55, 56]. Transposition is

a positive map, but it is not a completely positive map. For all bipartite states of

dimension 2× 2 and 2× 3 it has been proven that, a state is separable if and only if

it is positive under partial transposition operation,

ρ is separable → ρTA ≥ 0

This is the celebrated Peres-Horodecki criterion. It is a necessary and sufficient con-

dition for 2× 2 and 2× 3 systems.

22


2.14.2 The von Neumann entropy

The von Neumann entropy is a generalization of Shannon entropy for quantum systems

and it is a fundamental measure of quantum information. The von Neumann entropy

of the quantum system ρ is defined as,

S(ρ) = −Tr(ρlogρ) (2.50)

Let {λi} be the eigenvalues of the density matrix ρ, then the von Neumann entropy

can be redefined from the eigenvalues of the density matrix as,

S(ρ) = −∑i

λilogλi (2.51)

It is always a non-negative function. For all the pure states, S(ρ) = 0. For a com-

pletely mixed state defined in a d-dimensional Hilbert space, this quantity takes the

value log d [51].

2.14.3 Entanglement

For pure two qubit systems, entanglement is a von Neumann entropy of either of the

sub-systems,

E(|ψ〉) = −Tr(ρAlogρA) = −Tr(ρBlogρB)

where ρA and ρB are the sub-systems of ρ. Given n pairs of the two qubit systems

|ψ〉 distributed between Alice and Bob, it has E(|ψ〉) ebits of entanglement if n pairs

can be converted into m pairs of singlet states with 100% fidelity

Consider n pairs of the two qubit systems |ψ〉 is shared between two parties Alice

and Bob. Through the LOCC operation with the 100% fidelity m pairs is converted

irreversibly into the singlet state |ψ+〉 = 1√2

(|HV 〉 − |V H〉). For large value of n, the

ratio mn

gives the maximal number of singlet states available per copy and approaches

23

Entanglement detection and quantification Chapter 2

E(|ψ〉). Therefore, the system |ψ〉 is said to have E “ebits” of entanglement. The

process is known as entanglement distillation. Distillable entanglement gives the in-

formation about the amount of entanglement resource available in a quantum systems

[57, 58].

2.14.4 Entanglement of formation

Quantification of entanglement for a mixed state is a difficult problem. Let ρ be the

general mixed state of two quantum systems A and B. It is the statistical mixture of

pure states |ψi〉 with probabilities pi, which can be decomposed in innumerable ways

{|ψi〉|pi} as,

ρ =∑i

pi|ψi〉〈ψi| (2.52)

The entanglement of formation is defined as minimum of an average value of entan-

glement over all possible decompositions [59],

E(ρ) = min{|ψi〉|pi}

∑i

piE (|ψi〉) (2.53)

The entanglement of formation of ρ is zero if and only if ρ can be expressed as the

statistical mixture of product state, ρ =∑i

piρ(i)A ⊗ ρ

(i)B .

2.14.5 Concurrence

For arbitrary two qubit mixed states, Wootters [60] has derived an entangled measure

called concurrence, given by the expression:

C(ρ) = max{

0,√λ1 −

√λ2 −

√λ3 −

√λ4

}(2.54)

where the λ′s are the eigenvalues of R = ρρ and ρ is the spin flipped state defined

as:

ρ = (σy ⊗ σy)ρ∗ (σy ⊗ σy) (2.55)

24


Concurrence is an entanglement monotone viz., a quantity which is invariant under

local quantum operations and classical communication. For pure state |ψ〉, the con-

currence is defined as,

C(|ψ〉) =√ρρ = 〈ψ|ψ〉 (2.56)

2.14.6 Linear entropy

Linear entropy is a measure of mixedness of a quantum systems. Let ρ be an element

of the state space S(HN) of dimension N , then the linear entropy of ρ is given by

[61, 62],

S(ρ) =

√N

N − 1(1− Tr (ρ2)) (2.57)

Linear entropy S(ρ) = 0 for the pure states and S(ρ) = 1 for the completely mixed

states. For single qubit systems, linear entropy is given by, S(ρ) =√

2 (1− Tr (ρ2)).

For two qubit pure states |ψAB〉, the linear entropy of either of the sub-systems is

equal to the concurrence, that is

C (|ψAB〉) = S(ρA) (2.58)

2.15 Summary

The basics concepts in quantum mechanics and quantum information theory were re-

viewed in this chapter. In the ensuing one, a version of discrete Wigner functions given

by Wootters and Gibbons et al., will be presented as a prelude to the presentation of

the central results of the present thesis.

25

Chapter 3

Introduction to quasi-probability

distribution functions

This chapter provides a concise introduction to the phase space formulation of quan-

tum mechanics. A brief review of different discrete phase space constructions and their

applications is presented. The discrete Wigner function formulation given by Woot-

ters’ and Gibbons et al., is discussed in detail and the advantages and limitations of

this construction are also mentioned.

3.1 Introduction

The development of the phase space formalism of quantum mechanics has a rich his-

tory. Early investigators like Eugene Wigner were interested in developing a phase

space representation of quantum states in order to understand quantum systems in

their classical limit [9]. The motivation was to develop an alternate formalism of quan-

tum mechanics without reference to the wavefunction or the Schrodinger equation. In

such a formulation, quantum mechanics may be regarded as purely a statistical the-

ory. It was hoped that such an approach would help understand the foundations of

27

Introduction to Wigner Functions Chapter 3

quantum theory and its classical limits in the spirit of the correspondence principle.

Indeed, such descriptions are useful from different perspectives such as understanding

the emergence of classicality [5], derivation of semi-classical results [6], study of deco-

herence [5] and stochastic dynamics [7]. As early as 1932, Wigner had worked on the

representation of quantum mechanics in terms of joint quasi-probability distributions

of the position and momentum variables. Given the state of a quantum system |ψ〉, the

mean value of any function f(q,p) can be calculated. In general, there exists no joint

probability distribution W (q, p) such that 〈ψ|f(q,p) |ψ〉 =´ ´

W (q, p)f(q, p)dpdq.

All the same, it is possible to define functions of the form W (q, p) to satisfy the ear-

lier equation with suitable ordering of the operators and some other constraints. In

fact, innumerable such distribution functions can be defined [4, 8–12]. The Wigner

function, in particular, is extensively used in quantum optics as stated in the in-

troduction. In the context of modern developments in quantum computation and

quantum information studies, quantum states in finite dimensional Hilbert spaces has

gained widespread attention. This has lead to the development of discrete analogues

of Wigner functions. The present thesis focuses attention on one version of the discrete

Wigner function developed by Wootters and Gibbons et al [31–34]. The present chap-

ter provides a brief introduction to continuous and discrete Wigner functions. The

Wootters’ construction is discussed in substantial detail to facilitate the presentation

of the present work

3.2 Introduction to Wigner Functions

In classical statistical physics, if we know the phase space distribution wcl(q, p) then

we can calculate the average value of a function Acl(q, p), by

〈Acl(q, p)〉 =

ˆ ∞−∞

dq

ˆ ∞−∞

dpAcl(q, p)wcl(q, p) (3.1)

A pure quantum state can be represented as a wave function ψ(q) defined by ψ(q) =

28

Chapter 3: Quasi-probability distribution functions Chapter 3

〈q|ψ〉. The probability distribution in position can be identified from the wave function

by P (q) = |ψ(q)|2. If we know the wave function ψ(q) in position basis q, we can

calculate the wave function φ(p) in momentum basis through the Fourier Transform,

φ(p) =1√2π~

ˆ ∞−∞

exp(−ipq/~)ψ(q)dq (3.2)

The probability distribution in momentum space is therefore given by |φ(p)|2. We

may note here that the wave function gives a probability distribution as a function

of position or momentum but does not yield a joint probability distribution function

of both these variables. Such a joint probability distribution of both position and

momentum cannot be defined for quantum systems. The basic problem arises from the

fact that the Heisenberg uncertainty principle makes the simultaneous measurement

of position and momentum impossible and as a result, there are no trajectories in the

conventional interpretation of quantum mechanics. We may however note, that this

is not the case for the Bohm interpretation. The connection between an ensemble of

classical trajectories and quantum mechanics is established through the expectation

values of operators. In quantum mechanics, the expectation value of an observable is

given by 〈A〉 = Tr(ρA). The connection with a phase space distribution is established

by requiring 〈A〉 =´ ´

P (q, p)A(q, p)dqdp with P (q, p) ≥ 0 and´ ´

P (q, p)dqdp = 1.

A major impediment arises from the fact that the correspondence

A(q,p) −→ A(q, p) (3.3)

is not unique. Thus, for these reasons, valid probability distribution functions such as

P (q, p) do not exist. However, for the phase-space formulation of quantum mechanics

to be complete, a clear prescription for computing expectation values is essential.

Wigner derived such distributions functions by relaxing the condition P (q, p) ≥ 0 has

to be relaxed. Thus, these distribution functions are not valid probability distribution

29


functions and are hence called quasiprobability distributions. To understand Wigner

functions, it is first useful to consider the Weyl transformation which helps convert an

operator A(q,p) to a function of dynamical variable q, p . In the following sections,

we shall closely follow the exposition by Case [63] and Schleich [4] to formally introduce

Wigner functions.

3.2.1 Average of an Operator and the Weyl Transform

The correspondence between functions of phase space variables A(q, p) and the oper-

ator A(q,p) is provided by the Weyl transform,

A(q, p) =

êxp(−ipξ/~)〈q +

1

2ξ|A(q,p)|q − 1

2ξ〉dξ (3.4)

The Weyl Transformation of an operator A(q,p) in the momentum basis is given by,

A(q, p) =

êxp(−iqu/~)〈p+

1

2u|A(q,p)|p− 1

2u〉du (3.5)

Given a distribution function W (q, p), the expectation value of an operator A(q,p)

is given by,

〈A(q,p)〉 =

ˆdq

ˆdpA(q, p)W (q, p) (3.6)

Consequently,

Tr(AB) =1

h

ˆ Â(q, p)B(q, p)dqdp (3.7)

Therefore, the Weyl transform allows us to calculate the expectation values of observ-

ables by associating operators with functions.

3.2.2 Wigner functions

As discussed in chapter II, a pure quantum state can be represented by % = |ψ〉〈ψ|,

which can be expressed in the position basis as 〈q|p∣∣q′⟩ = ψ(q)ψ∗(q). This expression

can be readily generalized to mixed states. Using the properties of the Weyl transform

30


for product of two operators given above, we may write,

Tr[%A] =1

h

ˆ ˆρ(q, p)A(q, p)dqdp (3.8)

We may now define the Wigner function for pure states as,

W (q, p) = ρ/h =1

h

ˆe−ipξ/~ψ(q + ξ/2)ψ∗(q − ξ/2)dξ (3.9)

For a general mixed state , the Wigner distribution is generally given by,

W (q, p) =1

h

ˆ ∞−∞

exp

(−ipξ

~

)〈q +

1

2ξ|ρ|q − 1

2ξ〉dξ (3.10)

Given a decomposition of a mixed state {pi|ρi = |ψi〉〈ψi|} with Wi(q, p) being the

Wigner function of pure states |ψi〉, the Wigner function for mixed state ρ is given

by,

W (q, p) =∑i

piWi(q, p) (3.11)

Now, the expectation value of an operator may be written as,

〈A〉 =

ˆ ˆW (q, p)A(q, p)dqdp (3.12)

The Wigner distribution function has the important property that marginal distribu-

tions with respect to one variable can be found easily by integrating over the other

variable. The position distribution is given by,

w(q) = 〈q|ρ|q〉 =

ˆ ∞−∞

W (q, p)dp (3.13)

Similarly, the momentum distribution can be obtained by integrating the Wigner

31


function over the position,

w(p) =1

h

ˆ ∞−∞

dq′ˆ ∞−∞

dq”exp[−ip(q′ − q”)/~]〈q”|ρ|q′〉 (3.14)

Let W1(q, p) and W2(q, p) be the Wigner functions of the density operators ρ1 and

ρ2. The overlap between these two states ρ1 and ρ2 can be obtained from the Wigner

functions as,

Tr(ρ1ρ2) = 2π~ˆ ∞−∞

dq

ˆ ∞−∞

dpW1(q, p)W2(q, p) (3.15)

In the case of pure states, the transition probabilities can be obtained as,

|〈ψ1|ψ2〉|2 = 2π~ˆ ∞−∞

dq

ˆ ∞−∞

dpW1(q, p)W2(q, p). (3.16)

For orthogonal states, Tr(ρ1ρ2) = 0, from Eq (3.15), it is clear that,

ˆ ∞−∞

dq

ˆ ∞−∞

dpW1(q, p)W2(q, p) = 0 (3.17)

This imposes the condition that either W1(q, p) or W2(q, p) must take on negative

values. The Wigner function is bounded, therefore it cannot take arbitrary large

values. For a normalized pure states the Wigner function cannot take values larger

than 1π~ , that is,

|W (q, p)| ≤ 1

π~(3.18)

Wigner functions have the following translational property: Let W (q, p) be the Wigner

function of a density operator ρ and ρ′ is obtained from ρ through a Unitary operation

U = ei(y1q−x1p)/~

ρ′ = ei(y1q−x1p)/~ρe−i(y1q−x1p)/~ (3.19)

Then, the Wigner distribution W ′(q, p) of the new state ρ′ is obtained just by trans-

32


lating the elements of W (q, p), as

W ′(q, p) = W (q − x1, p− y1) (3.20)

3.2.3 Wigner functions of the ground and first excited states

of a harmonic oscillator

In this section, we examine the Wigner function of the ground and first excited states

of a harmonic oscillator. The Hamiltonian of the harmonic oscillator can be written

as,

H =p2

2+mω2q2

2(3.21)

The wave functions for the ground and first excited states of harmonic oscillator is

given by,

ψ0(q) =1

4√π√a

exp(−q2/2a2) (3.22)

ψ1(q) =14√π

√2

a

q

aexp(−q2/2a2) (3.23)

Using equation (3.9) it is easy to verify that the Wigner functions for the ground and

first excited state of the harmonic oscillator are,

W0(q, p) =2

hexp

(−a

2p2

~2− q2

a2

)(3.24)

W1(q, p) =2

h

(−1 + 2

(ap~

)2

+ 2(qa

)2)

exp

(−a

2p2

~2− q2

a2

)(3.25)

Plots for ground and first excited states of a Harmonic oscillator is given in Fig (3.1)

& (3.2).

33


Figure 3.1 – Wigner function of the ground state of a quantum harmonic oscillator.

Figure 3.2 – Wigner function of the first excited state of a quantum harmonic oscil-lator.

Figure 3.3 – Illustration of interference effects due to the superposition of two dis-placed ground states centered at q = +b and q = −b.

34


Figure 3.4 – Wigner function of a quantum system, which is a statistical mixture oftwo coherent states centered at q = +b and q = −b.

As a further example, consider the effects of superposition and classical mixture of

quantum states in phase space. Consider a harmonic oscillator ground state centered

at q = +b, we label it as ψ0(q − b) and its wave function is given by,

ψ0(q − b) =1

4√π√a

exp(−(q − b)2/2a2) (3.26)

The Wigner function of this wavefunction is a Gaussian, peaks at q = +b. In case, the

quantum system under consideration is a superposition of two ground states centered

at q = +b and q = −b, then the wave function of the system is given by,

ψ = A [ψ0(q − b) + ψ0(q + b)] (3.27)

corresponding Wigner function is,

W (q, p) =1

h (1 + e−b2/a2)e−a

2p2/~2[e−(q−b)2/a2 + e−(q+b)2/a2 + 2e−q

2/a2cos(2bp/~)]

(3.28)

This Wigner function not only contains two Gaussian functions centered at q = +b

and q = −b, but also contains interference terms that oscillate between these two

35

Review of Discrete Wigner Functions Chapter 3

peaks as shown in Fig (3.3). The interference is clearly an effect of superposition.

Suppose if a given quantum system is a classical mixture of these two ground states,

then the Wigner function of the system takes the form,

W (q, p) =1

2[W0(q − b, p) +W0(q + b, p)] (3.29)

W (q, p) =1

he−a

2p2/~2[e−(q−b)2/a2 + e−(q+b)2/a2

](3.30)

This Wigner function contains only two Gaussian functions peaks at q = ±b as shown

in Fig (3.4).

3.3 Review of Discrete Wigner Functions

3.3.1 Brief survey of DWF constructions

The generalization of the Wigner function to finite dimensional systems started with

the work of Schwinger [64]. Discrete Wigner functions are defined over a two dimen-

sional grid of points. As indicated earlier, there are different constructions of DWF

and we shall now quickly survey them. For an arbitrary N dimensional system, N2

orthogonal unitary operators need to be defined. Schwinger established a method

to identify these operators. In 1974, Bout used these operators and provided a new

phase space formulation in quantum mechanics to study Bloch electrons in a solid.

In his formulation, the operators are represented through a Weyl transformation with

the quantum states represented as Wigner functions. This method is applicable to

odd dimensions and the phase space is an N × N array of points [65]. For spin-12

systems, Cohen and Scully [29] and Feynman [35], portrayed the spin-12

systems in

discrete version of phase space as constituted by 2× 2 array of points. In 1980, Han-

nay and Berry introduced a 2N × 2N phase space to represent finite dimensional

systems [66]. In 1987, Cohendet et al., developed the DWF scheme for odd integer

spin systems. In their construction, the discrete phase space is a N×N array of points

36


[37]. Different versions of N × N phase space for the spin systems were developed

by Wootters [32–34], Galetti and De Toledo Piza [30]. The discrete Wigner function

developed by Wootters is applicable to prime or prime power dimensional systems

because it is defined over a finite (Galois) field. Vaccaro and Pegg applied Wootters’

construction to the quantum optics by defining the Wigner function for number and

phase operators for single-mode field [67]. In 1995, Leonhardt extended the work of

Cohendet et al., to spin systems of arbitrary dimensions and applied it to the quan-

tum state reconstruction problem [68, 69]. Luis et al., introduced a discrete analog

of the Wigner function for arbitrary dimensions and used it to represent the number

and phase operators of a two-mode field [70]. Asplund et al., gave a direct recon-

struction procedure for the DWFs of quantum systems whose dimension is a product

of prime numbers [49]. Bianucci et al., represented both states and the evolution of

arbitrary finite dimensional systems in 2N × 2N phase space [38]. Using this version

of phase space, Miquel et al., studied the properties of phase space representations of

quantum algorithms such as the quantum Fourier transform and the Grover’s search

algorithms. The direct correspondence between quantum and classical evolutions in

phase space was examined by these authors [71]. Lopez and Paz investigated quan-

tum walks on a cycle using the DWF and provided some insights into the effects of

interference that makes the quantum and classical walks different [43]. They also

studied the influence of the environment on the quantum coin carried by the walker

and gave intuitive descriptions of the decoherence process in phase space. Wootters

developed his earlier versions of DWF further and along with Gibbons et al. provided

a MUB based construction of DWFs for prime power dimensional quantum systems

[31]. This construction is consistent with Euclidean geometry and notion of lines and

sets of parallel lines.

37

Review of Discrete Wigner Functions Chapter 3

3.3.2 Applications of DWFs

Discrete Wigner functions have been applied to problems in the fields of quantum

information and quantum computation. Though hardly exhaustive, we shall now dis-

cuss some of the applications reported in the literature. DWFs have been used in

quantum teleportation [41, 42], quantum computational speed-up [45] and quantum

random access codes [72] by several investigators. It is a very hard problem in general

to establish the separability or otherwise of a given quantum state. A necessary and

sufficient condition exists only for density matrices of dimension 2⊗ 2 or 2⊗ 3. There

was therefore a natural interest in finding out whether DWFs can used to address

the separability problem. Pittenger and Rubin have given a DWF for prime power

dimensional systems and analyzed the separability of the system directly in terms of

the DWF [73]. Franco and Penna have characterized the entanglement properties of

two qubit DWFs. They observed that the two qubit DWF is entangled if and only if

either W or W PT have negative entries, where W PT is DWF corresponding to partial

transposed density matrix [74]. Ryu et al., have shown that the quasi-probability

distribution function becomes positive semi-definite if the expectation values of mea-

surements are described by the local hidden-variable model. They also proposed a

marginal distribution function and have shown that it can be used as an entanglement

witness for two qudit systems [75]. Bjork et al., have discussed the relationship be-

tween the translationally covariant DWFs and different versions of MUBs with respect

to their entanglement properties. They have shown that there exist three different

constructions of DWF in three qubit systems [76]. Klimov et al., have examined

the phase space structure for a system of n-qubits and have given a comprehensive

phase space approach to the construction of MUBS for n-qubits [77]. Another area

of interest pertains to quantum state reconstruction through an ensemble of projec-

tive measurements. Mutually Unbiased Basis sets can be used to reconstruct DWFs.

Adamson et al., provided the first experimental procedure to reconstruct the quantum

38


state of two photon polarization states using MUB measurements. This approach fa-

cilitates the direct reconstruction of DWFs of two qubit systems [50]. Klimov et al.,

have investigated the non-uniqueness of the DWF representation for odd and even

dimensions [78]. Another important application of DWFs has been in the area of

quantum computation. Cormick et al., and Gross have investigated pure states which

have non-negative DWFs called stabilizer states [40, 79]. They have shown that non-

negativity is preserved by the set of unitary operators which forms a subgroup of

the Clifford group. This result can be viewed as the discrete version of the Hudson’s

theorem. They have argued that the states which have non-negative DWFs and their

unitaries are classical in the sense that they can be simulated efficiently by classical

computers using the stabilizer formalism. For multiqubit DWFs, Cormick and Paz

showed that for coherent superposition of two stabilizer states, the interference fringes

may spread over all of phase space. They also found that for a subset of states, the

oscillatory region remains localized and these family of states could be used in quan-

tum error correcting codes [80]. Munoz et al., introduced discrete coherent states

for n-qubit systems in terms of the eigenstates of the finite Fourier transform and

derived a DWF for these states [81]. Casaccino et al., have computed an extrema

of the DWF using the complete set of phase-space point operators, and applied it in

the construction of quantum random access codes [72]. Veitch et al., have established

a connection between the negativity of the DWF of odd dimensions and quantum

computational speed-up [46]. Mari and Eisert have shown that for both continuous

as well discrete Wigner functions of odd prime dimensions, positive Wigner functions

can be simulated efficiently using only classical resources [82].

39

Gibbons et al., construction over a finite field Chapter 3

3.4 Gibbons et al., construction over a finite field

3.4.1 Galois fields

Field: Definition: A set F in conjunction with two laws of composition (+,×) is said

to be a field if it satisfies the following axioms,

1. The set F together with the addition operation “+” is an abelian group F+, and

its identity element is 0.

2. The set F − {0} together with the multiplication operation “×” is an abelian

group, F× = F − {0} and its identity element is 1.

3. Addition distributes over multiplication: a.(b+ c) = a.b+ a.c for all a, b, c ∈ F .

The set of real and complex numbers are natural examples of a field. The number of

elements in the field is defined as the order of the field. A field with finite number

of elements is called a finite field. Such a field exists if and only if the order of the

field is prime or power of prime. Finite fields are also called Galois field and denoted

by FN , where N is the order of the field. Two fields F and F ′ of the same order are

isomorphic to each other. For every finite field FN , there exists two positive integers

m and n with the condition m < n, such thatm∑i=1

1 =n∑i=1

1 and hence,n−m∑i=1

1 = 0. That

is there must exist a smallest positive integer λ, such thatλ∑i=1

1 = 0. The integer λ is

called the characteristic of the field. The characteristic of the field should be either

zero or prime. To prove this, let us assume λ = kl, then

λ∑i=1

1 = 0 (3.31)

(k∑i=1

1

)(l∑

i=1

1

)=

kl∑i=1

1 = 0 (3.32)

40


This implies eitherk∑i=1

1 = 0 orl∑

i=1

1 = 0, which is a contradiction. Therefore, the

characteristic λ should be prime. For every prime number there exists a prime field.

We shall discuss the extension of the prime field to the prime power dimensional

case. Let r be the prime number and Fr be the prime field. We denote Fr[x] as the

polynomial with coefficients in Fr [83]. The extension of the finite field to prime power

dimension requires the use of an irreducible polynomial which is discussed below.

3.4.2 Irreducible polynomial

Definition: For a given prime field Fr and positive integer n, let a0+a1x+· · ·+anxn =

0, where ai ∈ Fr, is said to be an irreducible polynomial of order n, if no solution

exists within Fr.

For the general prime power N = rn, the finite field FN is generated by finding

the irreducible polynomial of order n. Let ω be one of the solutions of the irreducible

polynomial which is not an element of Fr then, the other powers of ω are also solutions

of this polynomial and the set{

0, 1, ω, ω2, · · · , ωN−2}

satisfies all the axioms of the

field. Therefore, the elements of FN are given by FN ={

0, 1, ω, ω2, · · · , ωN−2}

. The

finite field Frn may also be considered as a vector space of dimension n. We can now

define a basis for this field as a set of elements E = {e1, e2, ....en}, where ei ∈ FN ,

and with that one can express all the elements q of the field as q =∑i

qiei. Next, let

us define the trace for the field elements and dual basis.

3.4.3 Trace of the field element

Definition: Trace is a map between Frn to Fr, that is tr(α) : Frn → Fr, defined as,

tr(α) = α + αr + αr2

+ .....+ αr(n−1)

(3.33)

Therefore, trace is an element of the prime field, which satisfies tr(α + β) = tr(α) +

tr(β).

41

Mutually unbiased bases Chapter 3

Dual basis: For a given basis E = {e1, e2, ....en}, the dual basis E = {e1, e2, ....en} is

uniquely defined as,

tr(eiej) = δij (3.34)

3.5 Mutually unbiased bases

Consider the state of a general quantum system represented by the density matrix

ρ of dimension N × N . To completely determine the density matrix ρ we need to

specify N2−1 independent real numbers. Every complete measurement on the system

has N possible outcomes. If each measurement associated with a basis choice has N

distinct outcomes associated with N eigenstates, N −1 distinct outcome probabilities

can be measured. But the total probability sums to one and each measurement

gives N − 1 independent probabilities. Since the number of independent real entries

in the density matrix is N2 − 1 and each basis choice gives N − 1 of them, the

number of distinct measurement sets required to reconstruct the density matrix is

(N2 − 1)/(N − 1) = N + 1. But identifying these N + 1 basis sets is not arbitrary

because, each measurement set should give maximally distinct information about the

system. Let M1 and M2 be two distinct measurement sets and B1 ={|vij〉}

and

B2 ={|vi′j′〉

}be their respective eigenbases, where |vij〉 denotes the jth vector of the

ith basis. These two measurements M1 and M2 gives maximally distinct information

about the system, when

|〈vij|vi′

j′〉|2 =1

N(1− δii′) + δii′δjj′ (3.35)

The bases B1 and B2, which possess this property are called Mutually Unbiased Bases

(MUBs) and the measurements M1 and M2, whose eigenbasis are MUBs are called

Mutually Unbiased Measurements (MUMs) [32, 48]. Therefore, the reconstruction

of the state using an ensemble of identically prepared systems represented in N di-

42


mensional Hilbert space requires N + 1 MUBs but, their existence is not proved for

arbitrary dimensions. However, when N is prime or prime power, the existence of

N + 1 MUBs is guaranteed. This dimensionality constraint is closely related to the

order of the finite fields. It is however known that for the dimension other than prime,

the number of MUBs is not more than N + 1.

3.6 Discrete phase space for prime dimensions

For a system defined in a Hilbert space of dimension N , the discrete phase space

is an N × N array of points. Like the continuous phase space, the horizontal and

the vertical axes of the discrete phase space are associated with two non-commuting

observables. As an example, for the discrete phase space of dimension N = 2, the

axes are associated with Pauli operators σz and σx. In the continuous case, points

are labeled by an ordered pairs of real numbers (q, p). One of the obvious candidates

to label points in the discrete case is the set of natural numbers {0, 1, 2, 3, · · · , N − 1}

under arithmetic modulo-N . But this association introduces an additional problem in

defining lines in the discrete phase space in that, two non-parallel lines may intersect at

more than one point. This limitation arises because there is no unique multiplicative

inverse for this set and hence, it does not have the mathematical structure of a finite

field. Therefore, points in the discrete phase space are labeled by the elements of the

Galois field FN , defined earlier. This association limits the applicability of the discrete

phase space to specific dimensions. Since the finite fields exists only for the prime

or prime power dimensions, discrete phase space also exists only for such dimensions.

Wootters’ constructions has a nice analogy with continuous Wigner function in the

sense that the notions of Euclidean geometry can be carried over to this construction.

Of these, the notion of a line plays an important role. To begin with, we denote

a point in the discrete phase space as an ordered pair, (q, p) where q, p ∈ FN . A

set of N points in the discrete phase is called line if it satisfies the linear equation

43

Discrete phase space for prime dimensions Chapter 3

aq + bp = c, where a, b, c ∈ FN . Two parallel lines have same a, b but different value

of c. In view of the continuous case, a line should have following properties,

1. For given pair of points in the discrete phase space, there is exactly one line

containing both the points.

2. Two non-parallel lines intersect exactly at one point, that is, they share only

one common point.

3. For a point α which is not contained in the line λ, there is exactly one line

parallel to λ containing the point α.

For fixed values of a and b when c is varied over FN in the equation aq+ bp = c, a set

of N parallel lines called a striation is generated. There are exactly N(N + 1) lines in

the discrete phase space which can be grouped into N + 1 sets of parallel lines. The

point (0, 0) is called the origin and any line which contains the origin is called a ray.

Each striation contains exactly one ray. For fixed values of field elements x and y, if s

varies over the field elements FN , the set of N points (sa, sb) = s(a, b) form the rays

of each striation. Striations which are formed by the fixed points (0, 1) and (1, 0) are

called the vertical and horizontal striations respectively. For prime power dimensions,

there are N+1 striations in the discrete phase space and N+1 MUBS. Each striation

contains N lines analogously each MUB contains N basis vectors. To give lines and

striations of the discrete phase a physical interpretation, we can associate the lines of

the phase space with pure states or the rank one projectors. Let λij be the i-th line

of the j-th striation and Q(λij) = |λij〉〈λij| be the rank one projector associated with

that line, where |λij〉 is the i-th vector of the j-th MUB.

For a given prime number r, there exists a finite field called the prime field Fr =

{0, 1, · · · , r − 1}. Finite fields of the prime power dimensions are generated from the

solutions of the irreducible polynomial of order r, with prime field elements being the

coefficients of the polynomial. By defining a basis B = {a1, a2, ..., an} for the finite

44


field FN , every element of the finite field FN can be expressed as q =n∑i=1

qiai, where

the expansion coefficients qi are the elements of the prime field Fr. For example, let

F2 = {0, 1} be the prime field, then the elements of the field F4 are generated from the

irreducible polynomial of order 2, i.e. x2 +x+1 = 0. If ω be one of the solution of the

irreducible polynomial, it induces the other solution ω2 = ω+ 1. Therefore, the finite

field of dimension 4 would be given by F4 = {0, 1, ω, ω2}. Just like the continuous

case, one can define a set of N2 translations T(q,p) in discrete phase space. When it is

acting on a point (q1, p1) results in the point (q+ q1, p+ p1). Thus, the action of T(q,p)

on a line shifts each point in the line by an amount (q, p). The ray of the vertical

striation is given by the set of points{

(0, 0) , (0, 1) , (0, ω) , · · · ,(0, ωN−2

)}and Ts(0,1)

be the set of translations formed by these points, where s varies over all the Galois

field elements FN . This set of translations has the special property that the lines in

the vertical striations are invariant under the action of these translations. Similarly,

for every striation, there exists N−1 translation operators which leave the lines in the

striation invariant and those translation operators are formed by the points of the ray

of the corresponding striation. As discussed earlier, the continuous Wigner function is

translationally covariant. For DWFs to have a similar property, we associate unitary

operators T(q,p) to every translation operator T(q,p) on the discrete phase having the

following properties: (i) The law of composition of the translations in the phase space

should hold good for product of the unitary operators, i.e., TαTβ ≈ Tα+β. (ii) For

N = rn and q =∑i

qiei and p =∑i

pifi, where qi, pi ∈ Fr,

T(q,p) = T(q1,p1) ⊗ T(q2,p2) ⊗ · · · ⊗ T(qn,pn) (3.36)

where T(qi,pi) is an unitary translation operator acting on the i-th subsystem. As

mentioned earlier, for the individual systems of dimension r, the generalized Pauli

matrices X and Z can be associated with the unit horizontal and the vertical trans-

lations T(1,0) and T(0,1) respectively. Let {|0〉, |1〉, · · · , |r − 1〉} be the standard basis

45


for the single qubit systems, then the generalized Pauli matrices have the property

that,

X|k〉 = |k + 1〉 (3.37)

and

Z|k〉 = e2πik/r|k〉 (3.38)

and obey the commutation relation, ZX = ηXZ.

Based on the Eq (3.36), Eq (3.37) and Eq (3.38), unitary operators are then defined

upto a phase factor as,

T(q,p) = Xq1Zp1 ⊗· · · ⊗XqnZpn (3.39)

The only condition that is imposed on the rank one projectors associated with the

lines of the phase space is that of their translational covariance, which is ensured by

the equation,

Q(Tαλ) = TαQ(λ)T †α (3.40)

Let λ′ be the line obtained by translating λ by an translation operator Tα. Corre-

spondingly, Q(λ′) is equivalent to shifting Q(λ) by Tα. As we have discussed earlier,

the ray of every striation is the set of points s(x, y), for the fixed values x and y of the

field and s varying over the all the elements of FN . This ray and the other lines of its

striation are invariant under the action of Ts(x,y). Translational covariance condition

is achieved if Ts(x,y) mutually commute with themselves as well as with Q(λ), for all

values of s. These Q(λ)’s are all simultaneous eigenstates of Ts(x,y). For any two

values s and t, Ts(x,y) and Tt(x,y) commute with each other if and only if,

n∑j=1

(sx)j(ty)j =n∑j=1

(tx)j(sy)j (3.41)

46


This condition is achieved if and only if the field basis E = {e1, e2, · · · , en} and

F = {f1, f2, · · · , fn} are related to each other by,

fi = ωei (3.42)

for all values of x, y, s and t, where ω is any element of the field. So, we have

the freedom to choose the field basis for the horizontal axis E but the translational

covariance condition uniquely fixes the field basis for the vertical axis as defined by

the Eq (3.42). Bandyopadhyay et al., have shown that for prime power dimensions,

one can find the N2 − 1 traceless, mutually orthogonal unitary matrices, which can

be partitioned into N + 1 subsets of equal size. The unitary matrices in each set have

the unique property of being mutually commuting. The simultaneous eigenvectors of

these operators forms a MUB set [84].

3.6.1 Discrete Wigner Functions

In this section, we derive the relationship between the density matrix and the DWF.

In doing so, we start with the fundamental property of discrete Wigner functions

that, its sum along a particular line λ is equal to the expectation value of the rank

one projector Q(λ) associated with that line. Now, the sum of the DWF elements

along this line is equal to the probability p(λ) = Tr [Q(λ)ρ] =∑α∈λ

Wα, where p(λ) is

the outcome of the projective measurement. So we can reconstruct the DWF of the

system by the N + 1 projective measurements. Next, for any given point α in the

discrete phase space, there are N + 1 lines which pass through it. Then, the sum of

Wigner elements S(α) along the N + 1 lines which contains the point α is given by,

S(α) =∑λ3α

∑β∈λ

Wβ (3.43)

47


and this can also be written as,

S(α) = NWα +∑γ

Wγ (3.44)

From Eq (3.43) and Eq (3.44) it is clear that,

NWα +∑γ

Wγ =∑λ3α

∑β∈λ

Wβ (3.45)

that is,

Wα =1

N

[∑λ3α

∑β∈λ

Wβ −∑γ

Wγ

](3.46)

where∑β∈λ

Wβ is probability to measure the systems along the projector Q(λ), that is∑β∈λ

Wβ = Tr [Q(λ)ρ] and∑γ

Wγ = 1 . Therefore, Eq (3.46) can be simplified as,

Wα =1

N

[∑λ3α

Tr [Q(λ)ρ]− 1

](3.47)

Wα =1

N

∑λ3α

Tr {[Q(λ)− I]ρ} (3.48)

Wα =1

NTr [Aαρ] (3.49)

where Aα’s are called the “phase space point operators”, associated with every point

of the discrete phase space, given by,

Aα =∑λ3α

Q(λ)− I (3.50)

The DWFs are defined by the Eq (3.49), that is for a given ρ we can find the DWF

associated with it. Using Eq (3.47), if we know the outcome of N(N + 1) projective

48


measurements p(λ) = Tr [Q(λ)ρ], the DWF of the systems can be reconstructed as,

Wα =1

N

[∑λ3α

p(λ)− 1

](3.51)

This is known as the Quantum State Tomography.

3.6.2 Properties of phase space point operators

1. Aα’s are Hermitian therefore their eigenvalues are all real.

2. Tr(Aα) = 1.

3. The phase space point operators are mutually orthogonal to each other, Tr(AαAβ) =

Nδαβ. Hence Aα’s form a basis for the set of N ×N matrices.

4. The sum of phase space point operators Aα along any line λ is equal to the

projectors associated with it, i.e., Q(λ) =∑λ3αAα.

5. Any density matrices can be expressed as,

ρ =∑α

WαAα (3.52)

where the expansion coefficients are the DWFs.

3.6.3 Properties of Discrete Wigner Functions

1. W ′αs are real.

2. Sum of the DWF elements along a line λ is equal to the outcome of the projection

operator Q(λ), that is p(λ) = Tr [Q(λ)ρ] =∑λ3α

Wα.

3. DWF is normalized-∑α

Wα = 1.

4. DWFs are translationally covariant: Let ρ and ρ′ are density matrices and they

are related to each other by the unitary translation operators by ρ′ = TβρT†β .

49


Let their corresponding DWFs be W and W ′ respectively. Then the DWF W ′

is obtained by merely shifting the DWF elements Wα by an amount β, that is

W ′α = Wα+β.

3.6.4 Labeling scheme for lines and striations for multiqubit

DWFs

For multiqubit systems, the discrete phase space is a N × N array of points, where

N = 2n. The N×N array of points of the multiqubit systems are labeled by the finite

field elements FN ={

0, 1, ω, ω1, · · · , ωN−2}

, where N = 2n and ω is the primitive root

of the irreducible polynomial of order n [31]. We now identify the origin by the pair of

field elements (0, 0). There exists exactly one line in each striation passing through the

origin called a ray. Here, the labeling convention for the striations would be as follows:

The rays of every striation can be formed by the set of points (sx, sy) = s(x, y), where

x, y are fixed elements of the field and s varies over all the field elements. As an

example, for the rays of the vertical and the horizontal striations, the fixed points are

(0, 1) and (1, 0) respectively. These rays are generated by the s(0, 1) and s(1, 0) where

s varies over all the field elements. Now the rays of the other N − 1 striations are

generated from the fixed points (1, ω), (1, ω2), · · · , (1, ωN−2). With this, we can label

the striations as given in table (3.1). We denote each ray by λ0i where the subscript

is the striation index and the superscript the line index.

Striation number Fixed point Rays of the striation1 (0, 1) λ0

1 ={

(0, 0), (0, 1), (0, ω), · · · , (0, ωN−2)}

2 (1, 0) λ02 =

{(0, 0), (1, 0), (ω, 0), · · · , (ωN−2, 0)

}3 (1, 1) λ0

3 ={

(0, 0), (1, 1), (ω, ω), · · · , (ωN−2, ωN−2)}

4 (1, ω) λ04 =

{(0, 0), (1, ω), (ω, ω2), · · · , (ωN−2, 1)

}...

......

N + 1 (1, ωN−2) λ0N+1 =

{(0, 0), (1, ωN−2), (ω, 1), · · · , (ωN−2, ω2N−4)

}Table 3.1 – Labeling scheme for striations

Now given the ray of the vertical striation λ01 the other lines of this striation can be

50


obtained using the set of N − 1 horizontal translation operators

{T(s,0) : s varies over the whole field FN

}

and the lines in this striation is given by λj1 = T(j,0)λ01 where j ∈ FN . Here we have

labeled the lines of the vertical striation as{λ0

1, λ11, · · · , λ

ωN−2

1

}. Instead of labeling

the lines with the field elements FN we can label them using integer by

{λ1

1 = λ01, λ

21 = λ1

1, λ31 = λω1 , · · · , λN1 = λ

ωN−2

1

}

. In Table (3.1), rays associated with each striations are given. Then other lines

in each striation (other than the first striations) are labeled using the set of vertical

translation operators{T(0,s) : s varies over the whole field FN

}, acting on the ray of

each striation. The same labeling convention can be used as in the case of labeling

the vertical striation. Therefore, with this convention one can label the lines and

striations of the phase space as λji , that is line-j of striation-i. Wootters’ construction

can be made transparent by way of considering it for simple systems. In the next

section this construction is illustrated for a single qubit system.

3.7 Discrete Wigner function of single qubit sys-

tems

The discrete phase space for the single qubit systems are an array of 2×2 points. The

horizontal and the vertical axes need to be associated with two non-commuting observ-

ables. Pauli matrices σx, σy and σz satisfy this criterion which are non-commuting

and have exactly two outcomes. We associate σz and σx operators with horizontal

and vertical axes respectively. The points in this 2× 2 phase space are labeled by an

elements of the Galois field F2 = {0, 1}, which is the prime field. There are 6 lines in

51

Discrete Wigner function of single qubit systems Chapter 3

phase space, these lines can be grouped into 3 sets of parallel lines (striations). For the

single qubit case there are 4 translation operators. The origin is the point (0, 0) and

there are three rays, one from each striation. The rays of vertical and the horizontal

striations are generated by the set of points s(0, 1) and s(1, 0) respectively and the

ray of the third striation is generated by s(1, 1). The lines in these three striations are

invariant under the action of translations Ts(0,1), Ts(1,0) and Ts(1,1) respectively. Every

line λ in the discrete phase space can be associated with the pure state represented by

the rank one projectors Q(λ). Every translationT(a,b) is effected by the corresponding

operator T(a,b). The basic horizontal and the vertical translation can be associated

with the Pauli operators T(0,1) = σz and T(1,0) = σx, therefore the general translation

operator T(q,p) can be written as, T(q,p) = σqzσpx. Then unitary operator associated with

the T(1,1) is given by, T(1,1) = σ1zσ

1x = iσy. These three operators are traceless and mu-

tually orthogonal. The eigenvectors of these operators forms a MUBs for 2-dimension.

The vertical and the horizontal lines are invariant under the translations T(0,1) and

T(1,0) respectively, therefore the pure state or the rank one projector associated with

these lines are eigenstates of the operators T(0,1) = σz and T(1,0) = σx. The eigenstates

of these operators are MUB and Fourier transform each other. The normalized eigen-

basis of T(0,1) = σz is the standard basis denote it by

|0〉 =

1

0

, |1〉 =

0

1

,

these states are associated with the vertical lines. The horizontal lines are associated

with the eigenstates of T(1,0) = σx are{|D〉 = 1√

2(|0〉+ |1〉) , |A〉 = 1√

2(|0〉 − |1〉)

}.

Eigenstates of T(1,1) = σy are{|R〉 = 1√

2(|0〉 − i|1〉) , |A〉 = 1√

2(|0〉+ i|1〉)

}associ-

ated with diagonal lines. But this association is not unique, there are 22+1 = 8

possible ways to do this assignment. Each possible association known as the Quan-

tum net. Once this association is made, we may define the phase space point operators

Aα to every point of the discrete phase space using Eq (3.50). For any 2× 2 density

matrix we can calculate the DWF using Eq (3.49).

52


Figure 3.5 – (a) Labeling the points of 2 × 2 phase space by a finite field F2. (b)Labeling the lines of the discrete phase space with pure states. (c) Lines and striationsof the 2 × 2 phase space, set of orange points are rays and the set of black points areother line of the striations.

3.8 Discrete Wigner function of two qubit systems

The discrete phase space of two qubit systems are the 4 × 4 array of points. As

mentioned earlier, the integer {0, 1, 2, 3} addition under modulo-4 cannot be used to

label the points in the discrete phase space because this set does not form a finite

field. This choice introduces some additional problems in defining the line in phase

space. Since the dimension of the Hilbert space is power of a prime number, that

is N = 22, we can extend the prime field F2 to the dimension 4. Let F2[x] be set

of polynomials with coefficients in F2. Then x2 + x + 1 = 0 is the only irreducible

polynomial in F2[x]. Let ω be one of the solutions of the irreducible polynomial, then

53

Discrete Wigner function of two qubit systems Chapter 3

the other solution is ω2 = ω+ 1. Therefore, the Galois field of dimension 4 is given by

F4 = {0, 1, ω, ω2}. The addition and the multiplication for F4 is given in Table (3.2).

+ 0 1 ω ω2

0 0 1 ω ω2

1 1 0 ω2 ωω ω ω2 0 1ω2 ω2 ω 1 0

× 0 1 ω ω2

0 0 0 0 01 0 1 ω ω2

ω 0 ω ω2 1ω2 0 ω2 1 ω

Table 3.2 – Addition and multiplication table for F4

Figure 3.6 – (a) Labeling discrete phase space points by elements of the finite fieldF4. (b) Labeling horizontal and vertical lines of the phase space by pure states.

Figure 3.7 – Lines and striations of 4× 4 discrete phase space.

Points in discrete phase space are labeled by elements of the Galois filed F4, as shown

in Fig (3.6). There are 5 sets of parallel lines, that is 5 striations in phase space

54


and each striation contains 4 lines. For every striation, there exists set of N − 1 = 3

translations, other than the trivial translation T(0,0), which leaves the lines in the

striation invariant. Translations associated with these five striations are Ts(0,1), Ts(1,0),

Ts(1,1), Ts(1,ω) and Ts(1,ω2), where s varies over F4. The first two translations are

associated with the vertical and the horizontal lines. As mentioned earlier, every

T(q,p) can be associated with the unitary operator T(q,p). Using Eq (3.39), we can

define the general unitary operator up to a phase factor as,

T(q,p) = Xq1Zp1 ⊗Xq2Zp2 (3.53)

There are 15 unitary translation operators along with the trivial unitary translation

operator T(0,0) = I. These 15 unitary translation operators can be equally partitioned

in to 5 sets and each set contains 3 operators. Therefore, with each striation there ex-

ists 3 traceless, mutually orthogonal operators, which mutually commute each other.

The common eigenstates of these operators are associated with the lines of the corre-

sponding striation. There exists 5 bases which are the common eigenstates of unitary

translation operators associated with each set. As mentioned earlier, these 5 bases are

mutually unbiased with respect to each other. The basis vectors associated with each

striation is given in Table (3.3). For example vertical lines are invariant under Ts(0,1)

and the corresponding unitary translation operators are {I ⊗Z,Z ⊗ I,Z ⊗Z}. The

common eigenstates of these operators are the standard basis vectors. We have free-

dom to assign any vector in this basis to vertical ray λ11. Once it is fixed, then the state

vector associated with other lines of the striation is uniquely identified by applying

the unitary translation operators Ts(0,1) on the state associated with the vertical ray.

Similar assignments can be used for the lines of the other striations. Each possible

assignment is called quantum net and there are 44+1 = 1024 quantum nets in 2-qubit

case.

55

Discrete Wigner function of two qubit systems Chapter 3

Striation Fixed points Unitary operators Basis associated with the striation

1 (0, 1) Z ⊗ I, I ⊗ Z & Z ⊗ Z

1000

,

0100

,

0010

,

0001

2 (1, 0) X ⊗ I, I ⊗ X & X ⊗ X 1

2

1111

, 12

1−11−1

, 12

11−1−1

, 12

1−1−11

3 (1, 1) Y ⊗ I, I ⊗ Y & Y ⊗ Y 1

2

1−ii1

, 12

1ii−1

, 12

1−i−i−1

, 12

1i−i1

4 (1, ω) X ⊗ Y , Y ⊗ Z & Z ⊗ X 1

2

11i−i

, 12

1−1ii

, 12

11−ii

, 12

1−1−i−i

5 (1, ω2) Y ⊗ X, Z ⊗ Y & X ⊗ Z 1

2

1−i1i

, 12

1i1−i

, 12

1−i−1−i

, 12

1i−1i

Table 3.3 – The mutually unbiased basis vectors associated with lines of the 4 × 4phase space. This table provides a fixed point to generate the ray associated with eachstriation and also the unitary operators that leave the lines of the striation invariant.

3.8.1 Equivalence classes of quantum nets

Figure 3.8 – Figure shows two possible equivalence classes available in a 2× 2 phasespace and each equivalence class contains four quantum nets. Within the equivalenceclass, two distinct quantum nets are related to each other by unitary transformations{σx,σy,σz}.

56


Two quantum nets Q and Q′ are said to be equivalent if and only if the projection

operators Q(λ) and Q′(λ), associated with every line λ, are related through a unitary

operator U . That is, there exists a unitary operator U such that for every line λ of

the phase space, Q′(λ) = UQ(λ)U †. For example, for the N = 2 case, if lines in

the vertical and the horizontal striations are associated with the eigenstates of the

Pauli’s σz and σx operators, the diagonal lines would end-up being the eigenstates

of the operator σy. Now, by assigning the states |H〉 and |D〉 = 1√2

(|H〉+ |V 〉)

to the rays of the vertical and the horizontal striations, the assignment of the state

|R〉 = 1√2

(|H〉+ i|V 〉) or |L〉 = 1√2

(|H〉 − i|V 〉) with the diagonal ray results in

two different equivalence classes. That is, the two quantum nets are not related

through any unitary operator U , it shown in Fig (3.8) Thus, there are 2 equivalence

classes in 2 × 2 phase space, and each equivalence class contains 4 quantum nets.

Generalizing this result, a system of dimension N has NN−1 equivalence classes, where

each equivalence class contains N2 quantum nets in all. The number of equivalence

classes for N = 4 is 64. Of these, only two of them have the special property that

the phase space point operators are tensor products of the A′αs of the single qubit

sub-systems. These operators take the form,

Aα = A1α1⊗ A2

α2(3.54)

and

Aα = A1α1⊗A2

α2(3.55)

where α = (q, p) and q, p ∈ F4. Finite field elements q and p can be expressed

as q = q1e1 + q2e2 and p = p1f1 + p2f2, where {e1, e2} and {f1, f2} are the finite

field basis for the horizontal and the vertical axes and qi, pi ∈ F2. The phase space

points of the individual systems are, α1 = (q1, p1) and α2 = (q2, p2). For these two

quantum nets, the projectors associated with each line are complex conjugates of each

57

Advantages and limitations of the Gibbons et al. construction Chapter 3

other. Since, each equivalence class contains N2 elements, there are only 32 quantum

nets having this special property. Finally, it would be useful to summarize some of

the properties of the Wootters’ construction and justify why we have take-up this

particular construction for further development.

3.9 Advantages and limitations of the Gibbons et

al. construction

As described in the introduction, there are several alternate formulations of Discrete

Wigner Functions, each having certain advantages over the other. Since the thesis de-

velops on the construction by Gibbons et al., some of the merits and disadvantages of

this construction warrant mentioning. In this construction, coordinates of the discrete

phase space are labeled by elements of the Galois field, therefore this procedure is ap-

plicable only for the prime power dimensional case. The resultant discrete phase space

is a d× d array of points, having several geometrical similarities with the continuous

phase space. It is possible to define the notion of line and sets of parallel lines called

striations in discrete phase space. The number of striations in the discrete phase space

matches with the maximum number of mutually unbiased basis (MUB). To give some

physical insights to lines of each striations, quantum states from the MUB sets are

mapped to these lines. This construction has the merit that DWFs have the transpar-

ent association as the expectation values of the phase space point operators. These

phase space point operators are constructed from the quantum states associated with

the lines. The expression Wα = 1NTr [Aαρ] can be readily inverted and the density

matrix can be written ρ =∑α

WαAα. Thus the W ′αs are no more than the expansion

coefficients of the density operator written in the basis of phase-space point operators

Aα. The disadvantage of this scheme is that each the DWF associated with a density

operator is not unique and there are NN+1 versions of it.

58

Chapter 4

Spin flip of multiqubit states indiscrete phase space

Time reversal and spin flip are discrete symmetry operations of substantial import to

quantum information and quantum computation. Spin flip arises in the context of

separability, quantification of entanglement and the construction of Universal NOT

gates. The present work investigates the relationship between the quantum state of a

multiqubit system represented by the Discrete Wigner Function (DWFs) and its spin-

flipped counterpart. The two are shown to be related through a Hadamard matrix that

is independent of the choice of the quantum net used for the tomographic reconstruction

of the DWF. These results would be of interest to cases involving the direct tomographic

reconstruction of the DWF from experimental data and in the analysis of entanglement

related properties purely in terms of the Discrete Wigner function.

4.1 Introduction

Spin flip and time reversal are involution symmetry operations which frequently arise

in quantum information studies. These are anti-unitary operations which are physi-

cally unrealizable in an idealized sense [85]. However, such operations are critically

important for entanglement detection, quantum optics, quantum computation and in

59

Introduction Chapter 4

the definition of entanglement measures [55, 56, 60, 86–89]. Phase space representation

of quantum states through Wigner functions provides a natural setting for understand-

ing time reversal. However, in the case of qubit systems, the relationship between the

Discrete Wigner function and its spin-flipped/time reversed counterpart is ill under-

stood, and the present work is an attempt at filling this gap. To appreciate how spin

flip arises in different contexts, the examples cited earlier are now considered in some

detail. It is generally a difficult problem to establish whether a mixed state is entan-

gled or separable. However, we recall that the celebrated Peres-Horodecki criterion,

based on positivity of partial transposition (PPT) [55, 56], provides the necessary and

sufficient condition for the separability of bipartite systems of dimension 2⊗2, 2⊗3 and

Gaussian states [86]. Partial transposition for a 2⊗2 system is defined by ρTij,kl = ρil,kj,

where ρ is the density matrix of the system. The connection between partial transpo-

sition and spin flip may be seen by writing partial transposition for two qubit separable

systems as:(a0 +−→a .−→σ

)⊗(b0 +

−→b .−→σ

)→(a0 +−→a .−→σ

)⊗(b0 +

−→b .−→σ

)−2bzσz, where

the σ′s are the Pauli operators. Thus, partial transposition amounts to a reflection

of the second particle through x-z plane. The connection with spin-flip is obtained by

effecting a π rotation after the reflection about the axis perpendicular to the x-z plane

[85]. On a single qubit, this operation is represented by the operator σyC, where C

is the complex conjugation of the state in the computational basis and σy, the usual

Pauli operator. In the language of quantum maps, ρ is separable if and only if selective

spin flip on the subsystem is a positive map. For a single qubit system represented

by the pure state vector |ψ〉, the spin flipped state is defined by |ψ >= (−iσy)|ψ∗ >

and likewise, for an arbitrary density matrix, it is defined as ρ = σyρ∗σy. Unitary

evolution constitutes a completely positive map which takes quantum states to quan-

tum states and in hindsight, it is not surprising that only anti-unitary operations are

effective for entanglement detection. In the context of quantum optics, the reflection

of the circularly polarized photon at a mirror, resulting in a orthogonal polarization

state is analogous to a spin-flip. This spin flip operation is essentially an inversion of

60

Chapter 4: Spin flip of multiqubit DWF Chapter 4

the Poincare sphere. The construction of Universal NOT (U-NOT) gates for selec-

tively flipping a single qubit is not perfectly feasible but its approximate realization

has been considered in the literature [90, 91]. Finally, since entanglement is viewed

as a resource, there is a strong requirement to quantify it. Several entanglement mea-

sures such as negativity, concurrence, tangles and their generalizations to qudit and

multiqubit systems have been proposed in the literature [60, 92–96]. The definition

of these measures critically hinge on the spin flip operation. For example, for arbi-

trary bipartite mixed states, Wootters [60] has derived an entangled measure called

concurrence, given by the expression:

C(ρ) = max{

0,√λ1 −

√λ2 −

√λ3 −

√λ4

}(4.1)

where the λ′s are the eigenvalues of R = ρρ and ρ is the spin flipped state defined

as:

ρ = (σy ⊗ σy)ρ∗ (σy ⊗ σy) (4.2)

Concurrence is an entanglement monotone viz., a quantity which is invariant under

local quantum operations and classical communication. Given the usefulness of this

quantity for entanglement quantification, there have been attempts at generalizing its

definition for systems of higher dimensions and for multipartite states [87, 97]. For

instance, the n-qubit concurrence Cn(ρ) is defined in terms of the eigenvalues of the

R = ρρ, where n-qubit spin flip operation is defined as:

ρ = σ⊗ny ρ∗σ⊗ny (4.3)

Though attempts at investigating entanglement have largely been based on represent-

ing the state through the density matrix, it is by no means the unique way of doing so.

In fact, alternate representations of the state through quasi-probability distributions

such as the Wigner functions and through Stokes vectors are prevalent in quantum

61


optics. Both these quantities can be readily reconstructed from measured data. These

quantities are therefore valid representations of the state in their own right. The rep-

resentation of the state through Wigner functions has the advantage that time reversal

has a very transparent interpretation. The extension of the Peres-Horodecki criterion

to bipartite Gaussian states by Simon [86] is based on the critical observation that

transposition in the continuous case is geometrically interpreted as a mirror reflection

in the phase space. This is evident from the observation that transformation ρ→ ρT

corresponds to the associated Wigner function transforming as: W (q, p)→ W (q,−p).

As discussed in chapter III, unlike continuous Wigner functions, Discrete Wigner

functions are not unique, and since the underlying field is discrete, certain restrictions

are imposed. In investigations involving the representation in terms of the DWF W ,

the spin flipped DWF W is required to qualitatively and quantitatively evaluate the

state. Given this background, there is strong motivation to examine the relationship

between W and W and to provide a prescription for the computation of the later from

the former. Towards this end, we exhibit an elegant relationship between W and W .

We show that the n qubit DWF is related to its spin flipped counterpart through an

N2 × N2 Hadamard matrix, where N = 2n. We show that the construction is inde-

pendent of the quantum net and supply the proof for a general n - qubit system. The

rest of the chapter is structured as follows: The spin flipped DWF is derived in two

steps in Section II. In section II-A, we show that W (∗), the DWF of ρ∗, is related to

W through a Hadamard matrix and II-B and we show that this is independent of the

choice quantum net. In II-C the DWF of the spin flipped density matrix is obtained

by effecting a generalized shift in W (∗). The final transformation matrix relating W

to W also turns out to be Hadamard matrix which is once again independent of the

quantum net. Section III-A illustrates the method for a single qubit system. Section

III-B outlines the derivation for the two qubit case. The final section provides the

conclusions with some brief remarks.

62


4.2 Spin flipped DWF of a multiqubit system

To derive a relationship between the DWF W and its spin-flipped counterpart W , we

begin by observing that the spin flip operation on the density matrix is a two step

process: the first step involves the complex conjugation of ρ in the computational

basis and the next one entails the application of the translational operators σ⊗ny to it.

Consequently, the computation of the spin flipped DWF W can be carried out in two

steps. We denote the DWF of ρ∗ by W (∗) (not to confused with complex conjugation

of W , which in any case is real valued). In the first step, we derive an expression for

W (∗) in terms of W . A shift associated with the translation σ⊗ny of ρ∗ is then effected

by subjecting W (∗) to a corresponding shift, to obtain W .

4.2.1 Derivation of W (∗) for the multiqubit state

If the system is represented by a state vector or a density matrix, then complex

conjugation is straight forward. The procedure is however not obvious when the

system is represented by the DWF. To elucidate this, we exploit the relationship

between W and ρ and write W(∗)α as:

W (∗)α =

1

NTr (ρ∗Aα) (4.4)

Now taking the complex conjugate of Eq (3.52) we have:

ρ∗ =∑β

WβA∗β (4.5)

substituting Eq.(4.4) in Eq. (4.5), thus:

W (∗)α =

1

NTr

[(∑β

WβA∗β

)Aα

](4.6)

63

Spin flipped DWF of a multiqubit system Chapter 4

taking the trace operation inside the summation:

W (∗)α =

1

N

∑β

WβTr(AαA

∗β

)(4.7)

There are N2 DWF elements in discrete phase space and these elements can be written

as a N2 × 1 column vector

W =(W(0,0), · · · ,W(0,ωN−2),W(1,0), · · · ,W(1,ωN−2), · · · ,W(ωN−2,ωN−2)

)T(4.8)

where first N elements of the DWF belong to the first line and so on. Similarly, in

the column vector notation the DWF W (∗) can be written as,

W (∗) =(W

(∗)(0,0), · · · ,W

(∗)(0,ωN−2),W

(∗)(1,0), · · · ,W

(∗)(1,ωN−2), · · · ,W

(∗)(ωN−2,ωN−2)

)T(4.9)

Therefore, Eq. (4.7) can be written in a compact form using the column vector

notation as,

W (∗) = FW

where F is a N2 ×N2 matrix, given by

F =1

N

Tr(A0,0A

∗0,0

)Tr(A0,0A

∗0,1

)· · · Tr

(A0,0A

∗ωN−2,ωN−2

)Tr(A0,1A

∗0,0

)Tr(A0,1A

∗0,1

)· · · Tr

(A0,1A

∗ωN−2,ωN−2

)...

......

...

Tr(AωN−2,ωN−2A∗0,0

)Tr(AωN−2,ωN−2A∗0,1

)· · · Tr

(AωN−2,ωN−2A∗ωN−2,ωN−2

)

(4.10)

We may consider this matrix as being made of N × N blocks, each of size N × N .

The Fij -th block is formed by the trace terms Tr(AαA∗β) by varying α over the

points in the line-i and β over the points in the line-j of the vertical striation, where

i, j ∈ {0, 1, 2, · · · , N}. Thus, Eq. (4.10) helps us implement the complex conjugation

operation for the DWFs. In the next section, we show that the matrix F is a Hadamard

64


matrix and that it is independent of the quantum net.

4.2.2 Proof that F is a Hadamard Matrix and that it is in-

dependent of the quantum net

While the DWF depends on the specific choice of the quantum net, we show that the

transformation matrix F is itself independent of this choice. Thus, it will be shown

that a single Hadamard matrix F is sufficient for transforming W , obtained using

any quantum net, to the corresponding W (∗). We now start by writing the explicit

equation for Tr(AαA∗β). Using Eq (3.50),

Tr(AαA∗β) =Tr

[(∑λ3α

Q(λ)− I

)(∑λ′3β

Q∗(λ′)− I

)]

=Tr

[∑λ3α

∑λ′3β

Q(λ)Q∗(λ′)

]−N − 2

=Tαβ −N − 2 (4.11)

where the term Tαβ = Tr

[∑λ3α

∑λ′3β

Q(λ)Q∗(λ′)

]contains (N + 1)2 terms of the form

Tr[Q(λji )Q∗(λj

′

i′ )]. The Q(λji )s are rank one projectors associated with the line-j of

the striation-i. In the N+1 MUBS, the first basis sets can always be taken to have real

elements and hence complex conjugation does not alter them. Therefore, for these

two cases, Tr[Q(λji )Q∗(λji )] = 1 (where i ∈ [1, 2]) as Q∗(λji ) = Q(λji ). The other

N−1 bases have complex entries and are closed under complex conjugation i.e,. com-

plex conjugation takes each basis vector to another one orthogonal to it. Therefore,

Tr[Q(λji )Q∗(λji )] = 0 for i ∈ [3, 4, . . . N + 1]. If however, Q(λji ) and Q∗(λj

′

i′ ) belong to

different striations i.e,. different bases sets, then, Tr[Q(λji )Q∗(λj

′

i′ )] = 1N

(i 6= i′). It

is important to note that for all these cases, Tr[Q(λji )Q∗(λj

′

i )] is independent of the

quantum net. This feature essentially arises from the fact that DWF is translationally

covariant.

Consider the case α = β in Tr(AαA∗β), which are the set of all diagonal elements

65

Spin flipped DWF of a multiqubit system Chapter 4

of the matrix F . From Eq (4.11) Tr(AαA∗β) = Tαβ − N − 2 , where the term Tαβ

has (N + 1)2 trace terms. There are N(N + 1) trace terms Tr[Q(λji )Q∗(λj

′

i′ )] = 1N

for which i 6= i′, that is Q(λji ) andQ∗(λj′

i′ ) are from different striations, so their

value becomes N(N + 1) 1N

= N + 1. In the remaining N + 1 terms, the value of

Tr[Q(λji )Q∗(λji )] is 1 for the first two striations and 0 for the other N − 1 striations.

Therefore, Tr(AαA∗α) = 1 for all α, so that all the diagonal entries of the F matrix

are all equal to 1.

Next, consider the case in which α and β belong to the same line in the vertical

striation. This spans all the diagonal blocks of the F matrix. The variable Tαβ in

Tr(AαA∗β) is N(N+1) times 1

N, since Tr[Q(λji )Q

∗(λj′

i′ )] = 1N

, for which, i 6= i′. Since

α and β are from the same striation, for each value of α and β, there are two trace

terms which contribute 1 and other terms are zero. So in this case Tr(AαA∗β) = 1.

Therefore, all the diagonal blocks of F are N ×N matrices with unit entries.

As the last case, consider α and β belonging to the different lines of the vertical

striation. This condition spans all the off-diagonal blocks of the F matrix. Here too,

the calculation of Tαβ gives N(N + 1) times 1N

for the i 6= j case. Since α and β are

from two distinct lines of the vertical striation Tr[Q(λj1)Q∗(λj′

1 )] = 0 ,∀α, β. For the

given α, if β runs over all the points in the given striation, it gives a particular row

in that block. In that row N2

entries are +1s and N2

are −1s.

Thus, in F , all the diagonal blocks are N ×N matrices having entries which are +1

and all the off-diagonal blocks are the N×N matrix with each row containing an equal

number of +1s and −1s. Further, the rows and columns of F are orthogonal to each

other. We thus see that F is a Hadamard matrix. We note that these observations

are not specific to any quantum net, implying that F is independent of the same.

4.2.3 The spin flipped Wigner function W

The next step now is to obtain the spin flipped DWF W from W (∗). In terms of

density matrices, the spin flip operation is defined as ρ = σ⊗ny ρ∗σ⊗ny . To find W , the

66


Pauli operators acting on the individual qubits are effectively translation operators

Tβ acting on W (∗), that shift each element of W (∗) in the phase space by an amount β.

We can realize this transformation by a N2 ×N2 matrix T acting on W (∗) (arranged

as a column vector) to obtain W :

W = TW (∗) (4.12)

T depends on the basis choice of the underlying Galois field. Using the Eq (4.10) we

can write W directly in terms of W as:

W = TFW (4.13)

T acting on F merely interchanges the rows, and so the resultant matrix is again a

Hadamard matrix H = TF . Thus, the spin flipped state takes the form:

W = HW (4.14)

This completes the derivation of W from W and the proof that the Hadamard matrix

H is quantum net independent.

4.3 Illustration of the spin flip operation for one

and two qubit DWFs

4.3.1 Illustration for a single qubit system

In order to clarify the procedure for computing W , let us consider the spin flip oper-

ation on a single qubit system. For this case, the axes of the discrete phase space are

labelled by the Galois field elements F2 = {0, 1}. In the 2× 2 phase space, there are

3 striations and each striation contains two lines, having totally 6 lines. The MUBs

67

Illustration of the spin flip operation for one and two qubit DWFs Chapter 4

associated with the 3 striations are,

B1 =

1 0

0 1

B2 = 1√2

1 1

1 −1

B3 =1√2

1 1

−i i

where the matrices are the basis sets and the columns of Bi are the mutually or-

thogonal basis vectors. The first two basis sets B1 and B2 are not altered by complex

conjugation, but for the third basis set B3, complex conjugation interchanges columns

1 and 2. From Eq (4.10) we have W (∗) = FW . F can be therefore be written as,

F =1

2

Tr(A00A∗00) Tr(A00A

∗01) Tr(A00A

∗10) Tr(A00A

∗11)


∗01) Tr(A01A

∗10) Tr(A01A

∗11)


∗01) Tr(A10A

∗10) Tr(A10A

∗11)


∗01) Tr(A11A

∗10) Tr(A11A

∗11)

=

1

2

F11 F12

F21 F22

(4.15)

where F is written as a block matrix with 2 × 2 blocks, each block being a 2 × 2

matrix. By examining Eq (4.11), clearly, Tr(AαA∗β) = Tαβ − 4 and,

• For points α = β, i.e for the diagonal entries of F11 and F22, Tr[Q(λji )Q∗(λji )] =

1 for i = 1, 2 and Tr[Q(λji )Q∗(λji )] = 0 for i = 3. Therefore, Tαα = [2(2 +

1)12] + 1 + 1 = 5. So, Tr(AαA

∗α) = 1. Therefore, for all the diagonal entries

Tr(AαA∗α) = 1.

• If α and β belong to the same line of the vertical striation then, Tr[Q(λj1)Q∗(λj1)] =

1, whenever Tr[Q(λj2)Q∗(λj2)] = 1 then Tr[Q(λj3)Q∗(λj3)] = 0 and vice versa.

Therefore, Tαβ = [2(2 + 1)12] + 1 + 1 = 5. So Tr(AαA

∗β) = 1. This argument

holds for all the diagonal blocks. Thus, the diagonal blocks F11 and F22 are of

the form:

1 1

1 1

.

• For the off-diagonal blocks, α and β belong to different lines of the vertical

striation and Tr[Q(λj1)Q∗(λj′

1 )] = 0 for all α and β. If points α and β belong

68


to the same line of the striation-3, then, Tr[Q(λj3)Q∗(λj3)] = 0. Therefore,

Tαβ = [2(2 + 1)12] = 3. So Tr(AαA

∗β) = −1, otherwise Tr(AαA

∗β) = 1. For

example α = (0, 0) and β = (1, 1) belong to different lines of the striation-1 and

same line of the striation-3, so Tr(AαA∗β) = −1.

The transformation matrix F thus takes the form,

F =1

2

1 1 1 −1

1 1 −1 1

1 −1 1 1

−1 1 1 1

(4.16)

In the construction of this matrix, no particular quantum net is used hence, F is

independent of the same. The matrix F has the following properties:

1. The elements of the F are either ± 1N

, where N = 2n.

2. Two different rows(columns) are orthogonal to each other.

3. Determinant of the F is −1. For a general n qubit system, the determinant of

F is equal to (−1)n.

4. It is self-inverse.

therefore, the DWF of the complex conjugated state can be written as,

W(∗)00

W(∗)01

W(∗)10

W(∗)11

=

1

2

1 1 1 −1

1 1 −1 1

1 −1 1 1

−1 1 1 1

W00

W01

W10

W11

(4.17)

The next task is to perform the unitary translational operation on W (∗) to complete

the spin flip operation. In the equation ρ = σyρ ∗ σ†y, the Pauli’s operator σy is

69


essentially a translational operator Tβ = σy, which translates every point in the

DWF W (∗) by an amount β. The spin flipped state is hence given by Wα = W(∗)α+β.

From Eq (3.39), it is seen that the general translational operator on phase space can

be written as T(q,p) = σqxσpz . From this equation, σy may be expressed as Tβ = σy =

−i(σ1xσ

1z) = T(1,1), where −i is the phase factor. Therefore, the spin flip operation is

carried out by translating every element of the phase space by an amount β = (1, 1).

This translation can also be represented by a matrix :

W00

W01

W10

W11

=

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

W(∗)00

W(∗)01

W(∗)10

W(∗)11

(4.18)

Therefore W may be written as W = TW (∗). We know that W (∗) = FW , hence using

Eq (4.17), and Eq (4.18), we may write,

W00

W01

W10

W11

=

1

2

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

1 1 1 −1

1 1 −1 1

1 −1 1 1

−1 1 1 1

W00

W01

W10

W11

(4.19)

or

W = HW (4.20)

Thus, the product H = TF , is also a Hadamard matrix as explained earlier.

4.3.2 Spin flipped DWF of a two qubit system

To appreciate the general results obtained for the multiqubit state, it would be helpful

to additionally consider the two qubit case in some detail. The points in the axis of

two qubit discrete phase space are labelled by the finite field F4 = {0, 1, ω, ω}. The

70


MUBS associated with this discrete phase space is given by:

B1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

B2 = 1

2

1 1 1 1

1 −1 1 −1

1 1 −1 −1

1 −1 −1 1

B3 =1

2

1 1 1 1

−i i −i i

i i −i −i

1 −1 −1 1

B4 = 1

2

1 1 1 1

1 −1 1 −1

i i −i −i

−i i i −i

B5 =1

2

1 1 1 1

−i i −i i

1 1 −1 −1

i −i −i i

As in the single qubit case, each matrix is a basis set and the columns are the basis

vectors. Since the first two basis sets do not have any complex entries, complex

conjugation does not alter them. But the last three bases have complex entries and

complex conjugation takes a vector into some other vector in the same basis. For two

qubit systems, the discrete phase space is a 4 × 4 array of points having 16 entries,

writing W as a 16× 1 column vector and using Eq (4.10) we have:

W (∗) = FW

where F is a 16 × 16 matrix. This 16 × 16 matrix can be considered as constituted

of 4 × 4 blocks where each block is again a 4 × 4 matrix and denote the blocks by

Fi,j. The F11 block is constructed by running through the points of the line-1 in the

71


vertical striation i.e., (0, 0), (0, 1), (0, ω) and (0, ω). So, F11 is given by,

F11 =

Tr(A0,0A

∗0,0

)Tr(A0,0A

∗0,1

)Tr(A0,0A

∗0,ω

)Tr(A0,0A

∗0,ω

)Tr(A0,1A

∗0,0

)Tr(A0,1A

∗0,1

)Tr(A0,1A

∗0,ω

)Tr(A0,1A

∗0,ω

)Tr(A0,ωA

∗0,0

)Tr(A0,ωA

∗0,1

)Tr(A0,ωA

∗0,ω

)Tr(A0,ωA

∗0,ω

)Tr(A0,ωA

∗0,0

)Tr(A0,ωA

∗0,1

)Tr(A0,ωA

∗0,ω

)Tr(A0,ωA

∗0,ω

)

(4.21)

In this block matrix, the value of Tr(AαA∗β) = Tαβ −N − 2., where,

Tαβ = Tr

[∑λ3α

∑λ′3β

Q(λ)Q∗(λ′)

](4.22)

In Tαβ, there are (4 + 1)2 trace terms, wherein Q(λji ) and Q∗(λj′

i′ ) are from different

striations, for which Tr[Q(λji )Q∗(λj

′

i′ )] is 14. In general, there are 4(4 + 1) such trace

terms in Tαβ . In the remaining (4 + 1) terms, Tr[Q(λji )Q∗(λj

′

i )] is either 0 or 1

depending on the points α and β. For example,

• if α = (0, 0) and β = (0, 0), Tr[Q(λji )Q∗(λj

′

i )] = 1 for i = 1, 2 and Tr[Q(λji )Q∗(λj

′

i )] =

0 for i = 3, 4, 5. Therefore Tαβ = [4(4 + 1)14] + 1 + 1 = 7. So Tr

(A0,0A

∗0,0

)= 1.

This argument holds for all the diagonal entries.

• if α = (0, 0) and β = (0, ω), Tr[Q(λji )Q∗(λj

′

i )] = 1 for i = 1, 3 and Tr[Q(λji )Q∗(λj

′

i )] =

0 for i = 2, 4, 5. Therefore Tαβ = [4(4 + 1)14] + 1 + 1 = 7. So Tr

(A0,0A

∗0,ω

)= 1.

Similar argument holds for all the off-diagonal entries of the diagonal blocks.

Therefore, all the diagonal blocks are 4×4 matrix whose entries are 1. Next, consider

one block from the off-diagonal blocks- F12 is one of them:

F12 =

Tr(A0,0A

∗1,0

)Tr(A0,0A

∗1,1

)Tr(A0,0A

∗1,ω

)Tr(A0,0A

∗1,ω

)Tr(A0,1A

∗1,0

)Tr(A0,1A

∗1,1

)Tr(A0,1A

∗1,ω

)Tr(A0,1A

∗1,ω

)Tr(A0,ωA

∗1,0

)Tr(A0,ωA

∗1,1

)Tr(A0,ωA

∗1,ω

)Tr(A0,ωA

∗1,ω

)Tr(A0,ωA

∗1,0

)Tr(A0,ωA

∗1,1

)Tr(A0,ωA

∗1,ω

)Tr(A0,ωA

∗1,ω

)

(4.23)

72


Here too, the crucial step is in calculating Tαβ. In Tαβ, there are 4(4 + 1) terms with

value 14. Since this block is formed by the lines-1 & 2 in the vertical striation, the

trace term Tr[Q(λj1)Q∗(λj′

1 )] = 0. For a given α, with β running over the points in the

line-2 in the vertical striation, 4 times the trace terms Tr[Q(λji )Q∗(λj

′

i )] becomes 1 for

i = 2, ..., 5. For α = (0, 0) and β = (1, 0), Tr[Q(λji )Q∗(λj

′

i )] = 1 for i = 2, 4 therefore,

Tr(A0,0A

∗1,0

)= 1 + 1 + (N + 1)−N − 2 = 1. Similarly, for α = (0, 0) and β = (1, ω),

Tr[Q(λji )Q∗(λj

′

i )] = 1 for i = 3, 5 therefore Tr(A0,0A

∗1,0

)= 1+1+(N+1)−N−2 = 1.

The remaining terms in that line are −1, that is Tr(A0,0A

∗1,1

)= −1 = Tr

(A0,0A

∗1,ω

).

Same arguments hold for the other columns of this block, that is the term Tr(AαA∗β)

becomes two times +1 and the other two times −1. So the block matrix takes the

form,

F12 =

1 −1 1 −1

−1 1 −1 1

1 −1 1 −1

−1 1 −1 1

(4.24)

Since Tr(A∗B) = Tr(AB∗), if A and B are Hermitian matrices, the block matrix

Fj,i = Fi,jT , where T stands for transposition . The columns of the off-diagonal blocks

have equal number of +1 and −1. So, the general transformation matrix has entries

which are either +1 or −1 and rows(columns) of F are orthogonal to each other.

Thus, the F is a Hadamard matrix of dimension 16 × 16. This transforms a DWF

(16× 1 column vector) to a complex conjugate DWF by,

W (∗) = FW

The second step, involves the calculation of W in terms of W (∗), where W is the DWF

of ρ . We may recall, that the spin flipped density matrix ρ = (σy ⊗ σy)ρ∗ (σy ⊗ σy)

as given by Eq (4.2). The effect of the translation operator Tβ = −σy ⊗ σy on ρ∗ is

to cause a shift in W(∗)α by an amount β, which depends on the basis choice for the

73


field. From Eq (3.39) the translational operator T(q,p)in discrete phase space can then

be written as,

T(q,p) = σq1x σp1z ⊗ σq2x σp2z (4.25)

Since σy = iσzσx, we can rewrite the translational operator Tβ = −σy ⊗ σy by:

T(q,p) = σxσz ⊗ σxσz (4.26)

From Eq (4.25) & Eq (4.26) it is clear that q1 = p1 = q2 = p2 = 1. If we choose (ω, 1)

as the basis of the horizontal axis, then the basis elements for the vertical axis also

turns out to be (ω, 1). Therefore q and p can be expressed in terms of the basis as:

q = q1.e1 + q2.e2 = 1.ω + 1.1

q = ω + 1 = ω (4.27)

and

p = p1.f1 + p2.f2

p = ω + 1 = ω (4.28)

So, for this particular choice of bases β = (ω, ω) and the translational operator Tβ =

−σy ⊗ σy = T(ω,ω). Therefore,

Wα = W(∗)α+β (4.29)

where β = (ω, ω).

74


W ∗ W

- - W ∗0,ω W ∗

1,ω W ∗ω,ω W ∗

ω,ω

- + W ∗0,ω W ∗

1,ω W ∗ω,ω W ∗

ω,ω

+ - W ∗0,1 W ∗

1,1 W ∗ω,1 W ∗

ω,1

+ + W ∗0,0 W ∗

1,0 W ∗ω,0 W ∗

ω,0

HH HV VH VV

−−−−−→σy ⊗ σy

- - W ∗0,ω W ∗

ω,0 W ∗1,0 W ∗

0,0

- + W ∗ω,1 W ∗

ω,1 W ∗1,1 W ∗

0,1

+ - W ∗ω,ω W ∗

ω,ω W ∗1,ω W ∗

0,ω

+ + W ∗ω,ω W ∗

ω,ω W ∗1,ω W ∗

0,ω

HH HV VH VV

Table 4.1 – The DWF of W ∗ subjected to rigid translation effected by σy⊗σy, resultsin the shifting of the elements of W ∗ by β = (ω, ω) to yield the corresponding elementof W .

Therefore, from Table 4.1, we see that, W can be calculated from W ∗ just by trans-

lating the elements of W ∗ by β = (ω, ω). If the Wigner elements are arranged as a

column vector then the this translation is carried out by,

W = TW (∗) (4.30)

But the DWF of the complex conjugated state is already a Hadamard transformation

of the original DWF, that is W (∗) = FW . So this Eq can be written as,

W = TFW (4.31)

W = HW (4.32)

It is fairly straight forward to show that the product of matrices T and F is again

a Hadamard matrix. With this, the underlying general method has been illustrated

using the one qubit and two qubit states as examples. From these illustrations, it

would be clear how the Hadamard matrix H may be pre-computed for any arbitrary

multiqubit state.

75

Conclusions Chapter 4

4.4 Conclusions

The experimental measurement of continuous Wigner function has been extensively

reported in the quantum optics literature. In these studies, quantum interference

effects are beautifully brought out and a link between non-classicality and negativity

of the Wigner function is transparent. However, unlike the continuous case, the

discrete Wigner function has not been as thoroughly investigated and barring some

examples, its utility is not all together clear. In the Gibbons’ et al. construction,

since the DWF depends on the choice of the quantum net, the non-classicality or

otherwise of the reconstructed state is not obvious. Evidently, a clear interpretation

of the consequences of spin flip would require the derivation of quantities that are

independent of the choice of the underlying quantum net. In the present work it has

shown that the DWF and the spin-flipped DWF of the multiqubit states are related

through a linear transform involving a Hadamard matrix. We have further shown that

this matrix is independent of the choice of the quantum net used in the reconstruction

of the DWF. The general method was illustrated for the one and two qubit discrete

Wigner functions. Experimentally, several protocols are available for the tomographic

reconstruction of the DWF but there are no entanglement measures defined purely in

terms of discrete Wigner elements. One way of defining entanglement measures for

DWFs is to use those defined for ρ and find equivalent expressions in terms of the DWF

elements. With the present results, we can readily compute bipartite concurrence in

terms of the DWF using the definition given in equation 4.1. With a bit of algebra, it

can be shown that the matrix R = ρρ may written as R = ρρ = 14

∑α

∑γ

WαWβAγAδ.

Likewise, one could also rewrite the expressions of the other tangles in terms of the

DWF. Alternately one may attempt to define altogether SLOCC invariant measures

starting from the DWF. The problem of deriving the equivalent of n-concurrence is

taken up for study in the next chapter.

76

Chapter 5

Stokes vector and its relationshipto DWF of multi-qubit systems

Stokes vectors and Discrete Wigner functions (DWF) provide two alternate ways

of representing the polarization state of multiqubit systems. A general relationship

between the Stokes vector and the DWF is derived for arbitrary n-qubit states for

all possible choices of quantum nets. The Stokes vector and the DWF are shown to

be related through a Hadamard Matrix. Using these results, a relationship between

the Stokes vector of a spin-flipped state and the DWF is derived. Finally, we also

present a method to express the Minkowskian squared norm of the Stokes vector,

corresponding to n-concurrence in terms of the DWF.

5.1 Introduction

In quantum optics, the quantum state of multiqubit systems can be variedly rep-

resented through the density matrix, Stokes vector and discrete Wigner functions

(DWFs). We have given a formal introduction to the density matrix and the DWF

in the previous chapters. The Stokes vector of the multiqubit systems will be defined

shortly. Of these three representations of the state, the density matrix is by far the

most widely used and techniques for entanglement detection and quantification are

77


defined in terms of this representation [53, 56, 60, 86, 96, 98]. However, Stokes vectors

and the DWFs are equally valid representations of the state and are both amenable

to direct measurements [49, 99–101]. Stokes vectors have the advantage that a en-

tanglement measure for multiqubit state called the generalized concurrence can be

defined[87, 88]. This measure is basically related to the Minkowski squared norm of

the Stokes vector.

The polarization state of a photon is represented by the Stokes vector with four

parameters related to the total intensity and the difference in the intensities associated

with measurements using three complementary basis sets. In quantum mechanical

terms, they are related to the expectation values of Pauli operators with respect

to the state ρ. For multiqubit systems, the Stokes parameters are the expectation

values of the generalized Pauli operators. Similarly, for a given multiqubit system, the

density matrix can be expanded in the basis of the generalized Pauli matrices and the

identity operator, where the expansion coefficients are entries in the Stokes vector.

Therefore, for multiqubit systems, there exists a linear transformation which connects

density matrix to the Stokes vector and vice versa. As seen earlier, DWFs are related

to probabilities associated with MUBs and these are expectation values associated

with the operators Aα’s. These phase space point operators form a basis for the

general multiqubit density matrices, and the expansion coefficients are the DWFs.

Therefore, the density matrix and the DWF can be calculated from the appropriate

linear transformation.

Thus, the density matrix and the DWF can be transformed into each other as can the

stokes vector and the density matrix. However, a direct transformation formula be-

tween DWF and the multiqubit Stokes vector is hitherto not available. This situation

is schematically illustrated in figure (5.1). In the present chapter, a direct relationship

between a general n-qubit Stokes vector and the DWF is derived, circumventing the

need to compute the density matrix as an intermediate step. In doing so, we need to

78

Chapter 5: Stokes vector and its relationship to DWF Chapter 5

address the fact that the representation of the state through the DWF is not unique,

though the Stokes vector is.

Figure 5.1 – Density matrix, multiqubit Stokes vector and the DWF are valid repre-sentations of the multiqubit systems. Figure shows the possible transformation formulasavailable to connect one representation to the other and the unavailability of the directrelationship between the DWF and the Stokes vector.

Conventionally, the computation of the Stokes vector of the given DWF involves some

messy algebra. To appreciate this fact, let us consider the definitions of the DWF and

Stokes vectors in terms of the density matrix : The density matrix ρ may be de-

fined by ρ =∑WαAα. It may be recalled that for polarization states of a n-photon

system, there are N2 phase space points, where N = 2n, and so the reconstruction

of ρ involves the addition of N2 matrices weighted by the DWF elements associated

with each point. Once the density matrix is constructed, the corresponding Stokes

vector is calculated using the expression Si1..in = 12nTr(ρσi1 ⊗ σi2 ⊗ ...⊗ σin). Since

the density matrix, DWF and the Stokes vector are all related through linear trans-

formations, this circuitous procedure can be avoided if a prescription is provided for

computing the Stokes vector parameters from the DWFs and vice versa. Earlier work

of M.Holmes and W. Schudy shows that for a given DWF, there exists a Hadamard

matrix which transforms it to the corresponding Stokes vector, but the form of the

Hadamard transformation for different quantum nets is not provided [52]. To the

best of our knowledge, the general prescription for obtaining this invertible transfor-

mation for a n-qubit system and its dependence on the quantum net are absent in

79

Stokes vectors and its relationship to density matrices Chapter 5

the literature. Since, for dimension N , there exists NN+1 possible quantum nets each

quantum net is related to a unique Hadamard matrix that transforms the DWF to the

Stokes vector. Since these Hadamard matrices are invertible, we show that the DWF

can be computed from the Stokes vector as well. Finally, we discuss some interesting

features associated with the spin flip operation. Bipartite Concurrence and its multi-

partite generalizations are important entanglement measures [60, 87, 88]. For n-qubit

systems, we define a family of Hadamard matrices SnH , such that for each Hadamard

matrix H, there exists a unique Hadamard matrix H that takes any given DWF to

the Stokes vector corresponding to the spin flipped state ρ.

This chapter is arranged as follows: Section II provides a short introduction to the

generalized Stokes vectors of the multiqubit systems. In section III-A, for trans-

parency, we illustrate our method of relating DWF to the Stokes for a single qubit

system and generalize the same for a multiqubit system in III-B. In III-C, we discuss

the procedure for obtaining the Stokes vector for the spin-flipped state. Derivation of

Minkowski squared norm in terms of the DWF is presented in Section IV. We conclude

the chapter in Section V with some brief remarks.

5.2 Stokes vectors and its relationship to density

matrices

5.2.1 For single qubit systems

Single qubit systems belong to a 2-dimensional Hilbert space and their Stokes vector

representation are four parameter column vectors. Consider the Pauli’s matrices

σx =

0 1

1 0

; σy =

0 −i

i 0

; σz =

1 0

0 −1

Pauli’s matrices satisfy the commutation relation [σi,σj] = 2iεijkσk. They are (i)

80


Hermitian, (ii) traceless Tr(σi) = 0 and (iii) mutually orthogonal to each other,

Tr(σiσj) = 2δij. Pauli’s matrices together with the identity operator σ0 = I2×2

forms a basis for the 2 × 2 Hermitian matrices. A general 2 × 2 density matrix can

be written in this basis as,

ρ =∑i

Siσi (5.1)

where i ∈ [0, x, y, z] and the expansion coefficients Si’s are elements of the column

vector S = (S0 S1 S2 S3)T known as the Stokes vector. This is an alternate represen-

tation of the quantum state. The Stokes parameters Si’s are the expectation values

of the Pauli’s matrices, given by,

Si =1

2Tr(ρσi) (5.2)

Since, the trace of the density matrix is one Tr(ρ) = 1, which uniquely fixes the

value of S0 = 12. For a general density matrix Tr(ρ2) ≤ 1, which implies that

4(S2x +S2

y +S2z ) ≤ 1. Pure states corresponds to 4(S2

x +S2y +S2

z ) = 1 and mixed states

corresponds to 4(S2x + S2

y + S2z ) < 1. Since, the parameter S0 is a constant always,

other three parameters uniquely define the state of the system. These three parameters

Sx, Sy and Sz can also be used as the coordinates for the three dimensional systems.

Then, any quantum system in the 2-dimensional Hilbert space, like polarization states

of photons, spin of electrons and in general any two level quantum system can be

represented as a point on or within the unit sphere. The pure states are on the surface

of the sphere while mixed states are the ones within the sphere and the completely

mixed one at the very center of the sphere.

5.2.2 For two qubit systems

For two qubit systems, the product operator basis σi ⊗ σj may be constructed,

where i, j ∈ [0, x, y, z]. Similar to the one qubit case, these operators are Hermi-

tian, traceless (except the operator σ0 ⊗ σ0) and mutually orthogonal to each other,

81

Stokes vectors and its relationship to density matrices Chapter 5

Tr [(σi ⊗ σj)(σk ⊗ σl)] = 4δikδjl. These operators forms a basis for the two qubit

density matrices. Any two qubit density matrix can be expanded in this basis as,

ρ =∑i,j

Sijσi ⊗ σj (5.3)

where Sij are the generalized Stokes parameters of the two qubit systems. Therefore

the general two qubit system can be expressed by these 16 real entries. The 16 × 1

column vector (S00 S10 S20 · · ·S33)T is known as the generalized Stokes vector of the

two qubit systems. The Stokes parameters are the expectation values of these product

Pauli’s operators,

Sij =1

4Tr(ρσi ⊗ σj) (5.4)

Since, Tr(ρ) = 1, the parameter S00 = 14

always. From the Eq (5.3) we can calculate

the reduced density matrices of the sub-systems,

ρA = TrB(ρ) = 2∑i

Si0σi (5.5)

and

ρB = TrA(ρ) = 2∑j

S0jσj (5.6)

where ρA and ρB are the reduced density matrices of the subsystems A and B. Let

us denote the Stokes vectors of the subsystems as SA and SB. These are related to

the generalized Stokes vector as, SAi = 2Si0 and SBi = 2S0i. With this, the Stokes

vector of the multiqubit systems may now be readily defined.

5.2.3 For multiqubit systems

The tensor product of the Pauli operators σi1 ⊗ σi2 ⊗ · · · ⊗ σin together with the

identity operator of dimension 2n× 2n act as the basis for the general density matrix.

The generalized Stokes vector of the multiqubit systems are the expectation values of

82


these product operators, given by,

Si1i2···in =1

2nTr (ρσi1 ⊗ σi2 ⊗ · · · ⊗ σin) (5.7)

Using the generalized Pauli operators and the identity operator the multiqubit density

matrix can be expressed as,

ρ =3∑

i1...in=0

Si1...inσi1 ⊗ σi2 ⊗ ...⊗ σin (5.8)

Similar to the two qubit case any kth subsystem can be written as,

ρk = TrA1···Ak−1Ak+1···An(ρ) = 2n−1∑i

Skjkσjk (5.9)

ρk = 2n−1∑i

S0···jK ···0σjk (5.10)

These Stokes parameters Si1i2···in uniquely define the state of the multiqubit system.

Using Eq (5.10), we can calculate the kth-subsystem of the multiqubit Stokes vector.

For the kth-subsystem the Stokes vector can be obtained from the multiqubit Stokes

vector as,

Skjk = 2n−1S0···jK ···0 (5.11)

5.3 Stokes vectors and its relationship to DWFs

5.3.1 Summary of known results for single qubits

The relationship between the Stokes parameter and DWF for a single qubit system

has been explicitly obtained by Holmes and Schudy for the Wootters’ net [52]. We

first summarize their results in the present section to facilitate a clear exposition of

our generalization to arbitrary multiqubit systems. Any one qubit system can be

83

Stokes vectors and its relationship to DWFs Chapter 5

represented using the 2× 2 identity matrix and the Pauli matrix as the basis

ρ = [s0I + ~s.~σ]

which may be written as,

ρ =3∑i=0

siσi (5.12)

Each element of the Stokes vector is given by,

si =1

2Tr(σiρ) (5.13)

Using the phase space point operators as the basis and DWFs as the weighting factor,

one can also express the density matrix as,

ρ =∑α

WαAα (5.14)

Thus, the Stokes vector and the DWF are characterized by four real parameters.

From Eq (5.12) and Eq (5.14), the difference between these two representations is

the following: Each element in the Stokes vector is reconstructed by the difference

in the intensities or the probabilities in three mutually unbiased basis sets. In the

case of the DWF, projective measurements yield only the sum of the DWF elements

associated with a line and not the individual entries. Hence, to obtain the value of a

single element, three projective measurements would be required.

For a single qubit, the horizontal and vertical axis are associated with Pauli’s σz and

σx operators respectively. The finite field elements F2 = {0, 1} are used to label the

points in this discrete 2× 2 “phase space”. Lines in the horizontal axis are associated

with the eigenvectors of the σz operators denoted by |H〉 and |V 〉. Lines in the

vertical axis are associated with the eigenvectors of the σx operators denoted by |D〉

and |A〉. Finally, the diagonal lines are associated with the eigenvectors of the σy

84


operators |R〉 and |L〉. The assignment of these eigenstates to the lines in the phase

space is not unique. Each possible assignment is a different quantum net. To facilitate

further analysis we now represent the set of Wigner elements {W00,W01,W10,W11}

by a column vector W = (W00,W01,W10,W11)T . Denoting the DWF of the Pauli

matrices σ0, σx, σy and σz by W I , WX , W Y and WZ respectively the DWFs of the

Pauli’s operators take the form shown in Table 5.1.

A 12

12

D 12

12

W I H V

A −12

−12

D 12

12

WX H V

A −12

12

D 12

−12

W Y H V

A 12

−12

D 12

−12

WZ H V

Table 5.1 – W I , WX , W Y and WZ are DWFs of the 2× 2 identity matrix and Paulimatrices σx, σy, σz respectively. The sum of the all the DWFs is related to the trace ofthe corresponding operator,

∑αW iα = Tr(σi), where i ∈ [I, x, y, z]. Therefore, the sum

of DWF elements of the W I is 2 and the sum is 0 for the DWF of the Pauli operators.

From Table 5.1, it is clear that each element of the DWF of the Pauli matrices is 12

multiplied by some phase factor. If U and V are the DWF of the two operators ρU

and ρV , then

Tr(ρUρV ) = 2∑α

UαVα (5.15)

Therefore, the Stokes vector S can be represented using this fact by,

S0 =∑α

W IαWα

Sx =∑α

WXα Wα (5.16)

Sy =∑α

W Yα Wα

Sz =∑α

WZαWα

85


Using the Eq (5.16)and the DWF of the Pauli operators from the Table 5.1, the Stokes

vector can be expressed as,

S0 =12

∑α

Wα

Sx =12

∑ij

(−1)jWij (5.17)

Sy =12

∑ij

(−1)iWij

Sz =12

∑ij

(−1)i⊕jWij

where ⊕ is addition modulo two. If the DWF is represented as a column vector, then

Eq (5.17) can be simplified as,

S0

Sx

Sy

Sz

=

1

2

1 1 1 1

1 −1 1 −1

1 −1 −1 1

1 1 −1 −1

W00

W01

W10

W11

(5.18)

S = HW (5.19)

where H is a constant times a Hadamard matrix. This Hadamard matrix depends

on the choice of the quantum net used to represent the Pauli operators as given in

table-5.1. For the one qubit system, there are 8 possible quantum nets. So that,

for each quantum net, there is one Hadamard matrix that takes the DWF to the

86


corresponding Stokes vector. The equation given above can be rewritten as,

S0

Sx

Sy

Sz

=

1

2

W00 +W01 +W10 +W11

W00 −W01 +W10 −W11

W00 −W01 −W10 +W11

W00 +W01 −W10 −W11

(5.20)

Since the sum of the Wigner elements along a line is associated with the probabilities

Eq (5.20) can be written as,

S0

Sx

Sy

Sz

=

1

2

1

P (+)− P (−)

P (R)− P (L)

P (H)− P (V )

(5.21)

This is a well known equation for reconstructing the general polarization state of

the photon using over-complete measurements. The phase factors in Eq (5.17) may

change for different quantum nets, however they result in the same probabilities.

This transformation given in Eq (5.18) is invertible and therefore the DWF is readily

constructed from the Stokes vector as,

W00

W01

W10

W11

=

1

2

1 1 1 1

1 −1 1 −1

1 1 −1 −1

1 −1 −1 1

S0

Sx

Sy

Sz

(5.22)

W = H−1S (5.23)

This step is crucial because, when we find the DWF associated with the given Stokes

vector, the quantum net used for the reconstruction of the DWF should be specified.

This information about the quantum net gets embedded in the form of the resulting

87


Hadamard matrix used for the transformation. In the next section, we generalize this

method to n-qubit systems.

5.3.2 Generalization to N-qubit systems

A general n-qubit state may be described using the generalized Pauli matrices as

basis,

ρ =3∑

i1...in=0

Si1...inσi1 ⊗ σi2 ⊗ ...⊗ σin (5.24)

and the n-qubit Stokes parameters can be calculated as,

Si1...in =1

2nTr(ρσi1 ⊗ σi2 ⊗ ...⊗ σin) (5.25)

Let W be the DWF of ρ and U i1...in be the DWF of the operator σi1 ⊗σi2 ⊗ ...⊗σin .

Then the Stokes parameters are directly computed from the DWFs of the n-photon

polarization state and the generalized Pauli matrices by,

Si1...in =∑α

WαUi1...inα (5.26)

The DWF elements of the generalized Pauli matrices are ± 12n

. Therefore from Eq

(5.26), the generalized Stokes parameters can be written using the DWF as

Si1...in =1

2n

∑α

(−1)f(α)Wα (5.27)

Here, the exponent f(α) for the general n-qubit case would obviously depend on the

choice of the quantum net. For a one qubit state, an explicit form for f(α) is provided

in Eq (5.17). While there is inherently no difficulty in its computation for a n-qubit

system, it must be explicitly calculated for a specific quantum net and a suitable field

basis. To clarify this point further, without going into the details of the steps involved,

we briefly describe the computation of f(α) for the Wootters’ net of a two qubit

88


system. For this case, the Stokes vector can be written as Si1i2 = Tr[(σi1 ⊗ σi2)ρ],

where i1, i2 ∈ [0, 1, 2, 3]. Let (ω, 1) be the field basis for the horizontal axis, which

choice, uniquely fixes the field basis for vertical axis to be (ω, 1). The point α = (q, p)

in the discrete phase space can then be expressed as q = q1.ω+q2.1 and p = p1.ω+q2.1,

where q,p ∈ F4 and q1, q2, p1, p2 ∈ F2. Here (q1, p1) and (q2, p2) are the phase space

points associated with the individual qubits. Consider the functions, fm(0) = 0,

fm(1) = pm, fm(3) = qm, f1(2) = q1 ⊕ p1 and f2(2) = q2 ⊕ p2, where (qm, pm) is the

phase space point of the m-th qubit. For the Wootters’ net, the Stokes vector of the

two qubit systems takes the form,

Si1i2 =∑(q,p)

(−1)f1(i1)⊕f2(i2)W(q,p) (5.28)

Thus, in the case of a N-qubit system, arranging the elements of the DWF as a column

vector, the corresponding Stokes vector can be calculated using the N2 ×N2 matrix

by,

S = HW (5.29)

Since, the product operators σi1 ⊗σi2 ⊗ ...⊗σin are orthogonal to each other, that is

Tr [(σi1 ⊗ σi2 ⊗ ...⊗ σin)(σj1 ⊗ σj2 ⊗ ...⊗ σjn)] = 2nδi1j1δi2j2 · · · δinjn

Two different rows (columns) are orthogonal to each other. Here, the N2×N2 dimen-

sional matrix is a Hadamard matrix is weighted by the factor 1N

, where N = 2n. As

in the single qubit case, the inverse of this Hadamard matrix transforms the Stokes

parameter to the corresponding DWF. For the n-qubit system, we now define the set

of all Hadamard matrices SnH containing NN+1 elements as,

SnH ={H(1), H(2), . . . , H(NN+1)

}(5.30)

89


where H(k) refers to the Hadamard matrix associated with the kth quantum net. In an

experimental context, once a choice of measurement basis is made for the tomographic

reconstruction of the state, the Hadamard matrices can be pre-computed and one can

go back and forth between the description of the state in terms of the DWF or the

Stokes vector.

5.3.3 Spin flip operation for n-qubit systems

Spin flip is an important symmetry operation in the fields quantum information and

quantum computation and results pertaining to this operation were presented in chap-

ter IV. It may be recalled that for a single qubit represented as a point on the Poincare

sphere, spin flip takes the point to one anti-podal to it. As explained earlier, this op-

eration being an involution symmetry operation involving complex conjugation, it

cannot be realized experimentally. Nevertheless it is an essential tool for entangle-

ment detection and its quantification. For multiqubit states, the spin flip operation

was defined as ρ = σ⊗ny ρ∗σ⊗ny , where the ∗ operation stands for complex conjugation

in the computation basis and σy the Pauli matrix. It was noted earlier that spin flip

is an antiunitary operation [85]. In chapter IV, we proved that the spin flip oper-

ation can be performed on a DWF of the multiqubit systems through a Hadamard

transformation which is independent of the quantum net. If W and W are the DWFs

(arranged as a column vector) of the state and the spin flipped state respectively, of

the n-qubit system, then W can be calculated from W by,

W = HW (5.31)

where H is the Hadamard matrix that is different from one that used to transform

DWF to the Stokes vector. Therefore, H does not belong to the set SnH . From Eq

90


(5.29) we can write the Stokes vector of the spin flipped state as,

S = HW (5.32)

Using Eq (5.31), one can directly calculate S from the given DWF W by,

S = HHW (5.33)

It is important to note here that, H ∈ SnH , but, H /∈ SnH . However, the product

HH is always the element from the set SnH . We denote this new element by H =

HH. Interestingly we find, that for a given H, H is obtained by flipping each state

associated with the quantum net Qi. Therefore in the set SH for every H, there exists

one unique H which transforms a DWF to its spin flipped Stokes vector.

5.4 Minkowsky squared norm of an N-Qubit state

in terms of the DWF

For n - qubit systems the Stokes parameters and the density operator are defined in

Eqs (5.7) and (5.8). Based on this, we may define the Stokes scalar as,

S2(n) =

1

2n[(S0...0)2−

n∑k=1

3∑ik=1

(S0...ik...0)2

−n∑

k,l=1

3∑ik,il=1

(S0..ik...il...0)2

− ...+ (−1)n3∑

i1,i2,....,in=1

(Si1....,in)2] (5.34)

For a n - photon polarization states this Stokes scalar is invariant under SLOCC and

it is a O0(1, 3) group invariant length [87]. This scalar quantity is also called the

Minkowskian squared norm. It is related to the corresponding density matrix ρ and

91

Minkowsky squared norm of an N-Qubit state in terms of the DWF Chapter 5

its spin flipped density matrix ρ by,

S2(n) = Tr(ρρ) = Tr(R)

where R = ρρ is used quantify the entanglement of the n - qubit systems S2(n). For

two qubit systems, concurrence can be calculated from the eigenvalues of R matrix.

Therefore the quantity S2(n) = Tr(ρρ) is very useful in calculating the n-concurrence

of the multiqubit system. Here, we show that we can calculate the n-concurrence

of the system, C2(|ψ >) = S2(n) directly for a given discrete Wigner function. To

compute the n-concurrence we use the fact, if ρ and σ are two different states and

W and V are the corresponding DWFs, then, Tr(ρσ) = N∑α

WαVα. So in this case,

the pure state concurrence can be written as C(|ψ >) =√Tr(ρρ) =

√N∑α

WαWα.

Using the column vector notation for the DWF, concurrence can be calculated as,

C(|ψ >) =√NW T W =

√NW T HW (5.35)

If W is the DWF of the pure two qubit system, then concurrence can be calculated

directly from the given two qubit DWF by the relation, C(|ψ >) = 2√W T HW .

The relation between the multipartite entanglement measure S2(n), mixedness of the

state M(ρ) and the measure of spin flip symmetry of the state is given by,

S2(n) +M(ρ) = I(ρ, ρ) (5.36)

where M(ρ) = 1 − Tr(ρ2) = 1 −∑α

W 2α and I(ρ, ρ) = 1 − D2

HS(ρ − ρ) can also be

defined as the measure of the indistinguishability of the state from its spin flipped

state, where D2HS(ρ− ρ) =

√12Tr [(ρ− ρ)2] is the Hilbert-Schmidt distance between

state and its spin flipped counterpart[88]. For a pure multiqubit states, M(ρ) = 0,

92


from Eq (5.35) and Eq (5.36) it is clear that,

S2(n) = I(ρ, ρ) = NW T HW (5.37)

Therefore, for a pure multiqubit systems the entanglement measure S2(n) and I(ρ, ρ)

are equal and this can be calculated directly from the multiqubit DWF using Eq

(5.37).

5.5 Conclusions

Each of the different representations of the quantum state of multiphoton systems

brings with it certain advantages. Though these representations are related through

linear transformations, the physical insights and computational advantages provided

by one is not readily translated in terms of the other. For continuous quantum systems,

the representation of the state by Wigner functions provides a clear-cut distinction

between classical and quantum states of light. The Wigner function for the former

are positive but the latter can be negative. Phase-space representations of such states

provide deep insights into quantum interference effects. In the case of discrete mul-

tiqubit systems, the relationship between different representations is little explored.

The representation of multiqubit system through density operators provides the tools

to distinguish between separable and entangled states atleast for pure states. Entan-

glement measures are also defined in terms of density operators. In optics, the Stokes

vector, both the classical and quantum versions, provides a direct experimental means

of measurement. The less prevalent DWF too has proved to be useful in the context

of quantum computation, stabilizer codes for error correction and so forth. Whenever

optical qubits are used in the case of quantum information or quantum computation,

it is useful to understand the relationship between these representations. The present

chapter is an attempt at examining such an inter-relationship. Here we have exhibited

the existence of a simple relationship between the DWF and the Stokes vector. The

93


two were shown to be related through a Hadamard matrix which can be computed for

any choice of quantum net used for the construction of the DWF. This prescription

for obtaining the complete set SH of Hadamard matrices associated with every choice

of the quantum net was provided. Thus, independent of the measurement context

under which data was obtained, the present results enable us to easily switch between

one representation and the other.

94

Chapter 6

Generalized reduction formula forDWFs of multi-qubit systems

For density matrices, the partial trace operation is used to obtain the quantum state of

subsystems, but an analogous prescription is not available for discrete Wigner Func-

tions. In the present work, we derive a reduction formula for discrete Wigner func-

tions of a general multiqubit state which works for arbitrary quantum nets. These

results would be useful for the analysis and classification of entangled states and the

study of decoherence purely in a discrete phase space setting and also in applications

to quantum computing.

6.1 Introduction

For a composite quantum system represented by a density operator, the state of the

sub-systems can be obtained through the partial trace operation. In the case of DWFs,

the reduction operation equivalent to the partial trace operation is not available in the

literature. However, the problem of performing reduction of the composite state DWF

to that of the sub-system has hitherto been addressed only for two qubit systems [52].

This result based on the requirement that the phase space point operators should

have a tensor product structure. In this work, the reduction formula is given only

95


for specific quantum nets called the Wootters and the Aravind nets. For a 4 × 4

phase space corresponding to a two qubit system, there are 1024 possible quantum

nets, but phase space point operators have a product structure only for 32 of them

and the reduction formula of Holmes et al., is applicable only to these cases [31]. For

other powers of prime, the existence of the product structure of the phase space point

operator has not been investigated. In any case, a reduction procedure for arbitrary

multiqubit systems is not known to the best of our knowledge. In the present work,

we derive such a generalized reduction formula, that does not require the existence

of such a product structure. In the earlier chapter we had addressed the problem of

carrying out spin flip operations on multiqubit DWFs and based on this result, we had

given a formula for quantifying the n-concurrence of the multiqubit systems directly

from the DWF. The relationship between the Stokes vector representation and DWF

for different choices of the quantum net was exploited for this purpose. In the present

work, we use some of the results obtained earlier, to provide a general method for

the reduction of the DWF to that of its subsystems. This prescription works for all

possible quantum nets of the global system as well as for those of the subsystems. The

current chapter is arranged as follows: Section II discuss earlier results obtained by M.

Holmes et al., which are important to the present work. In section III, the reduction

of single qubit DWF from two qubit DWF is presented for an arbitrary quantum net

as an illustrative example of the present approach. Section IV provides a derivation

of a general partial trace formula for multiqubit systems. Section V-A discusses the

relationship between Minkowskian squared norm and the spectrum of single qubit

DWFs and section V-B discuss the direct method of obtaining the entanglement of

pure two qubit DWFs. Sections VI sums up the relevance of the present work as

conclusions.

96

Chapter 6: Generalized reduction formula Chapter 6

6.2 Earlier results on two-qubit DWFs

Holmes et al., have given a method of performing the partial trace operation for two

qubit systems [52]. Their result is based on the product structure of the phase space

point operators given in the work by Gibbons et al., As mentioned in chapter II, for a

two qubit system, Aα’s have a product structure only for 32 quantum nets. Consider

the two qubit DWF defined in the quantum net, for which the phase space point

operator given by Aα = A1α1⊗ A2

α2. If this definition is used for the reconstruction

of the density matrix, it is easy to show that the density matrices of the subsystems

1 and 2 are,

ρA =∑α1

∑α2

Wα1,α2Aα1 (6.1)

and

ρB =∑α1

∑α2

Wα1,α2Aα2 (6.2)

respectively. It is clear from Eq (6.1) and Eq (6.2) that, the phase space point oper-

ators of the subsystems are complex conjugates of each other. That is, the DWF of

the subsystems 1 and 2 are defined in different quantum nets. The DWF of the first

subsystem can be calculated from Eq (6.1) and from definition of DWF as,

WAβ =

∑α2

Wβ,α2 (6.3)

Since, the DWF of the subsystem-2 is defined on a quantum net where the projection

operators associated with each line have been complex conjugated, it is necessary to

perform a spin flip operation along the y direction on the DWF of subsystemB i.e. WBβ

to obtain the correct state. This can be achieved by applying the Hadamard matrix

F defined in chapter-IV by, WB = FWB(∗). Alternatively, we shall now show that

this result can be achieved in the following manner: the phase space point operator

97

Earlier results on two-qubit DWFs Chapter 6

Aβ can be used in the place of Aβ in the equation WBβ = 1

2Tr(ρBAβ

)to obtain the

proper DWF. This is obvious from the fact ρB =∑α2

WB(∗)α2

Aα2 =∑α2

Wα2Aα2 , where

WB(∗)α2

is the DWF associated with ρ∗B. Hence, the DWF of the subsystem-2 takes the

form,

WBβ =

1

2Tr (ρBAβ) (6.4)

WBβ =

1

2Tr

[(∑α1

∑α2

Wα1,α2Aα2

)Aβ

](6.5)

WBβ =

1

2

∑α1

∑α2

Wα1,α2Tr(Aα2Aβ

)(6.6)

where the trace product Tr(AαAβ

)for two different point α = (qα, pα) and β =

(qβ, pβ) is given by Tr(AαAβ

)= (−1)(qα⊕qβ)(pα⊕pβ), where ⊕-is addition modulo-2.

Therefore, Eq (6.6) can be written as,

WBβ =

1

2

∑α1

∑α2

(−1)(qα2⊕qβ)(pα2⊕pβ) Wα1,α2 (6.7)

From Eq (6.3) and Eq (6.7), we can calculate the DWF of the subsystems for a given

two qubit DWF. For the other equivalence class, the complex conjugation operation

needs to be performed on the first subsystem rather than the second, with the DWF

of the second being defined on the chosen net. In this context, Gibbons et al., have

pointed out that the existence of the tensor product structure is itself not established

for other powers of prime. In the present work, we provide a reduction formula for

multiqubit DWFs where such a product structure is not required. To the best of our

knowledge, such a general result is not available in the literature.

98


6.3 A general reduction formula for two qubit DWFs

As shown in the earlier section, the approach by Holmes et al., is restricted to only

32 of the possible 1024 quantum nets. In this section, we derive a general result valid

for all quantum nets of the global as well as the subsystems. Let ρAB be the density

matrix of the two qubit system, ρA and ρB be those of its subsystems. In the density

matrix representation, the subsystem can be obtained by taking a partial trace on

ρAB i.e. ρA = TrB(ρAB) and ρB = TrA(ρAB). Now, to derive a formula for obtaining

the DWF of the single qubit subsystem from that of the two qubit DWF, we need to

specify the quantum nets of both. Hence, the transformation formula must be general

enough to accommodate this requirement.

Let M be the observable acting on the subsystem A of the general system ρAB. This

can be mathematically represented as (M ⊗ I)ρAB. The expectation value of the

observable M only on the subsystem ρA and the expectation value of the operator

M ⊗ I on the global system ρAB are one and the same, that is,

<M >ρA=<M ⊗ I >ρAB (6.8)

For general two qubit systems, the Stokes vector is a 16 parameter real valued column

vector, S = [S00, Sx0, Sy0, Sz0, S0x, S0y, S0z, · · ·Szz]T where the entries in the column

vector are the expectation values of the generalized two qubit Pauli matrices,

Si1i2 =1

4Tr(ρσi1 ⊗ σi2) (6.9)

where i1, i2 ∈ [0, x, y, z]. Replace the operator M in the Eq (6.8) with the Pauli’s

operators σi,

< σi >ρA=< σi ⊗ I >ρAB

These expectation values are essentially the Stokes vector of the first subsystem SA,

99

A general reduction formula for two qubit DWFs Chapter 6

given by

SAi = Tr[σiρA] = Tr[(σi ⊗ I)ρAB] = Si0 (6.10)

From Eq (6.10) it is clear that the Stokes vector of the first subsystem SA is part

of the two qubit Stokes vector SAB, that is, SAi = SABi0 . The Stokes vector of the

subsystem can be obtained from that of the Stokes vector of the global system by the

construction of the transformation matrix T1, such that

SA = T 1S (6.11)

where T 1 is the 4 × 16 matrix given by T 1 = 2[I OOO] with I is a 4 × 4 identity

matrix and O is a 4× 4 matrix with all entries being zero.

Let W be the DWF of the the system ρAB. From Eq (5.29), two qubit Stokes vector

S can be calculated from W by,

S = H2W (6.12)

where H2 is the 42 × 42 dimensional Hadamard matrix, which is an element of the

set SH2 , that is the set of Hadamard matrix for the two qubit systems. Based on the

quantum net of W , we can choose the Hadamard matrix from the set SH2 . Therefore,

from Eq (6.11) and Eq (6.12) one can calculate the Stokes vector of the first subsystem

directly from the DWF of the two qubit system by,

SA = T 1H2W (6.13)

where the product T 1H2 is the 4 × 16 matrix. The inverse transformation from the

Eq (5.23) takes the Stokes vector to the DWF of the subsystem A, by

WA = H−11 SA (6.14)

100


where H1 is the 4×4 Hadamard matrix, contained in the set SH1 of single qubit system.

Therefore from the Eq (6.13) and Eq (6.14), the DWF of the first subsystem can be

given by,

WA = H−11 T 1H2W

AB

that is

WA = P1WAB (6.15)

where P1 = H−11 T 1H2 is a 4 × 16 matrix. Using this relation, one can calculate

the DWF of the first subsystem from the DWF of the two qubit system. That is

Eq (6.15) performs the reduction operation for the two qubit DWF. By the similar

transformation, one can construct the reduction operation for the second subsystem

by a suitable construction of the matrix T 2 as,

WB = P2WAB (6.16)

where P2 = H−11 T2H2. Hence, the reduction operation for the general two qubit

DWFs can be performed using Eq (6.15) and Eq (6.16). Here, it is important to note

that, the reduction formula is general enough to compute the DWF of the subsystem

defined in any arbitrary quantum net from the DWF of the two qubit system defined

in an arbitrary quantum net. In our transformation equations, the information about

the quantum net of the global system and that of the subsystems are contained in

the Hadamard matrices H2 and H1. So equations (6.15) and (6.16) carries out the

reduction operation for chosen quantum net. The reduction formula given by Holmes

et al., in Eq (6.3) and Eq (6.7) are the special cases of this formula.

101

Reduction formula for the general multiqubit DWF Chapter 6

6.4 Reduction formula for the general multiqubit

DWF

In the density matrix formalism, from the given n-qubit state ρ, the state of an

arbitrary k-qubit subsystem can be calculated by the partial trace operation. In this

section, we derive a method of performing the equivalent of a partial trace operation on

the general n-qubit DWF by “tracing out”n−k-qubits . In the DWF setting, this can

be done using the following facts: the state of the sub-system can be readily extracted

from the Stokes vector of the composite state and the transformation formula between

Stokes vector and the DWF given in Eq (5.29) can be used to obtain the DWF of

the sub-system. To see how this may be accomplished, consider a density matrix of

the general n-qubit system ρ and let M i be some observable acting on the i-th qubit.

Given ρ, the expectation value of the observable M i on i-th qubit can be calculated

as:

<M i >ρi = Tr(M iρi) (6.17)

= Tr[(I ⊗ I ⊗ ...⊗M i ⊗ ...⊗ I)ρ] (6.18)

This can be generalized for any k-partite subsystem. When this problem is cast in

terms of Stokes vectors, the observables are the Pauli operators and the expectation

values < σj >ρi ’s are the Stokes parameters Sij of the subsystem i. That is, Sij =<

σj >ρi . But from Eq (6.18), it is clear that, < σj >ρi=< I ⊗ I ⊗ ...⊗σij ⊗ ...⊗ I >ρ.

This implies,

Sij = S0...j...0 (6.19)

Therefore, in the Stokes vector representation, the state of the i-th subsystem is a part

of the multiqubit Stokes vector S. Similarly, for any k-partite subsystem, its state

102


is contained in the multiqubit Stokes vector from which it can be easily extracted.

Consider the case of tracing out the last (n − 1) subsystems from the n-qubit state,

giving the state of the first system S1. The Stokes vector of the first system can be

calculated as,

S1 = T1S (6.20)

where T1 is the 4×4n matrix, given by T1 = 2n−1[I OO ... O] with I is a 4×4 identity

matrix and O is a 4 × 4 matrix with all entries being zero. Similarly any k-qubit

Stokes vector can be constructed from the Stokes vector of the multiqubit systems

with the help of a suitable transformation matrix Tk.

Sk = TkS (6.21)

where Tk is a 4k × 4n matrix. For a given multiqubit DWF W , the corresponding

Stokes vector can be calculated using the 4n × 4n Hadamard matrix Hn ∈ SHn by,

S = HnW (6.22)

where the Hadamard matrix Hn ∈ SHn . Let SHn be the set containing NN+1 possible

Hadamard matrices associated with each quantum net. Therefore, from Eqs (6.21)

and (6.22),

Sk = TkHnW (6.23)

where the subscript n of Hn indicate that the Hadamard matrix picked up from the

set SHn of the n-qubit systems. Eq (6.20) allows us to calculate the Stokes vector of the

k-qubit subsystem from the multiqubit DWF. Here, the knowledge of the quantum

net of W is implicitly available in the Hadamard matrix Hn. Using the inverse formula

103

Reduction formula for the general multiqubit DWF Chapter 6

given in Eq (5.23), we can find the DWF of the k-qubit system as,

W k = H−1k TkHnW (6.24)

where Hk ∈ SHk .

W k = PkW (6.25)

Therefore, Eq (6.25) helps us perform the reduction operation for the multiqubit

DWF. Thus, we find that the quantum net of the global system and the subsystems

are to be obtained from the Hadamard matrices Hn ∈ SHn and Hk ∈ SHk respectively.

Hence, the choice of the quantum net of the global system and that of the reduced

system can be freely made by an appropriate choice of the corresponding Hn and Hk.

6.4.1 Minkowskian squared norm and the spectrum of the

single qubit DWF

There exists a simple relatioship between the eigenvalues of the density matrix and

Minkowskian squared norm of the Stokes vector for a single qubit systems. This

relation helps us to calculate the eigenvalues of the density matrix ρ directly from

the corresponding DWF W . For a multiqubit systems we derived a n-concurrence

or Minkowskian squared norm of the Stokes vector of the system directly from the

DWF, by

S2n = Tr(ρρ) = W T HW (6.26)

where ρ is spin flipped state of the system, defined as ρ = σ⊗ny ρ∗σ⊗ny . For a single

qubit system, the quantity S21 acts as a measure of mixedness of the system. For

polarization states of the photon, this quantity is called the degree of polarization. S21

takes values between 0 (corresponds to pure states) and 1 (corresponds to completely

mixed states). In terms of the Stokes vector S, the Minkowskian squared norm is

104


given by,

S21 = S2

0 − S2x − S2

y − S2z (6.27)

where S0 is related to the trace of the density matrix, S0 = 12Tr(ρ) = 1

2and |r|2 =

S2x + S2

y + S2z is called the length of the Stokes vector. It is easy to show that,

S21 = det(ρ) (6.28)

S21 = λ1λ2 (6.29)

The eigenvalues of the density matrix is given by,

λ1/2 =1±

√1− 4S2

1

2(6.30)

Therefore, for a given DWF W the eigenvalues λ associated with its corresponding

density matrix ρ can be calculated directly from the DWF using Eq (6.30), where S21

can be calculated from eq (6.26).

6.4.2 Entanglement of two qubit pure states

In this section, we provide a direct method to calculate the entanglement present in

the DWF of two qubit pure states. The entanglement of two qubit pure states |ψAB〉

is given by the von Neumann entropy of the subsystems ρA or ρB [102], that is

E(|ψAB〉) = S(ρA) (6.31)

von Neumann entropy of the system ρ can be calculated as [51],

S(ρ) = −Tr(ρ lnρ) (6.32)

105


If λi’s are the eigenvalues of ρ, then

S(ρ) = −∑i

λi lnλi (6.33)

Let WAB be the DWF of the two qubit system. Using the partial trace formula

given in Eqs (6.15) and (6.16) we can calculate the DWF of the subsystems WA and

WB. For a single qubit DWFs W the Minkowskian squared norm can be given by,

S2n = W T HW . Eq (6.30) helps us to calculate the eigenvalues of the density matrix

ρ directly in terms of the DWF W . Therefore, using the partial trace formula and

Eq (6.33) one can calculate the entanglement of the pure two qubit DWF.

For a general quantum system represented in the d-dimensional Hilbert space, the

linear entropy can be calculated as [62],

SL(ρ) =d

d− 1

[1− Tr(ρ2)

](6.34)

The concurrence of the two qubit pure states |ψAB〉 are the linear entropy of either of

the subsystems, that is C2(|ψAB〉) = SL(ρA). The concurrence of the pure two qubit

DWF WAB is given by,

C(WAB) =

√√√√2

(1−

∑α

WA2

α

)(6.35)

Therefore, the entanglement and the concurrence of the two qubit DWFs can be

calculated using the reduction formula.

6.5 Conclusions

There are many contexts in the fields of quantum computation and quantum informa-

tion where access to state of the subsystem is vital. The quantification of entanglement

present in a composite bipartite system through Concurrence and the derivation of

106


monogamy relationships from tripartite entangled states are typical examples. Simi-

larly, in the case of multiqubit systems, the distribution of entanglement over suitably

partitioned subsystems is a problem of interest. Frequently, one is required to enlarge

the Hilbert space by taking a tensor product of the system with that of the envi-

ronment, subjecting the joint system to a unitary evolution and eventually tracing

out either the environment or the system. The theory of POVMs and weak measure-

ments are typical examples of such procedures. Hitherto, such techniques have been

uniquely applied to the case where the state of the system is represented in terms of

the density matrix and an equivalent approach was not available at least in the case

of systems represented by the discrete Wigner function. While the representation of

the state of continuous systems by Wigner functions has found widespread use, this

is not the case for the DWF due to some obvious limitation. An important limita-

tion with DWF as stated earlier has been the absence of a general reduction formula,

which problem has been addressed in the present work. While it is true that DWF,

density matrix and Stokes vector representations are but linear transforms of each

other, experimental situations could make one choice or the other more favorable and

experimental reconstructions of the different representation are also different. Going

by the experience with continuous system, where the phase space representation of

the state provides certain unique insights, further development of its discrete analog

is warranted. Motivated by such considerations, the present work is a step in the di-

rection of developing the relevant tools for the Discrete Wigner function of multiqubit

systems.

107

Chapter 7

Experimental reconstruction oftwo-qubit DWF

A simple experimental illustration of the construction of the DWF and the theoretical

results derived in the thesis are presented for entangled bipartite photonic states

generated through SPDC. Details of the experimental setup developed as a part of

this thesis is also presented here.

7.1 Introduction

In the earlier chapters, the theoretical procedure for obtaining the partial trace of a

multiqubit DWF, obtaining its spin flipped version and establishing the connection

between the Stokes vector and the different versions of the DWF were presented for

arbitrary multiqubit states. In the present chapter, we attempt to experimentally il-

lustrate these results for polarization entangled states produced through spontaneous

parametric downconversion (SPDC). A basic introduction to SPDC is provided along

with the details of the experimental set-up in sections II and III. In section IV, the

results of the tomographic reconstruction of the DWF using MUBS is presented. Ap-

plications of some of our theoretical results to the experimentally reconstructed two-

qubit DWF are then discussed. Fidelity and purity of these states are then computed

109

Spontaneous parametric down-conversion process Chapter 7

in terms of the DWF. In the concluding remarks, we highlight some shortcomings in

present analysis and indicate them as future work to be carried out.

7.2 Spontaneous parametric down-conversion pro-

cess

In linear optical processes, when a pump beam of frequency ωp is incident on a crystal,

the induced polarization inside the material depends linearly on the incident electric

field, which can be written as,

P = ε0χ(1)E (7.1)

where ε0 is permittivity of free space and χ(1) first order dielectric susceptibility, which

is a rank two tensor. In a non-linear optical process, the induced polarization inside

a crystal depends on the high order powers of the electric field and it is given by,

P (t) = ε0(χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + ...

)(7.2)

where χ(n) is a n-th order susceptibility, which is a tensor of rank n + 1. Induced

polarization inside the crystal can also be written as,

P (t) = p(1)(t) + p(2)(t) + p(3)(t) + ... (7.3)

where p(n)(t) is a n-th order polarization. The basic requirements for these nonlinear

processes are (1) high intensity laser source and (2) a bi-refringent crystal with non-

vanishing higher order susceptibility [103]. Second order susceptibility results in a

wave-mixing process, giving rise to new frequency components of the electromagnetic

radiation. Sum and difference frequency generation are some typical examples of such

processes. Spontaneous parametric down conversion (SPDC) is a second order non-

linear, three-wave mixing process and therefore occurs only in non-centrosymmetric

110

Chapter 7: Experimental reconstruction of two-qubit DWF Chapter 7

crystals [104]. When laser light of frequency ωp is incident on the downconversion

crystal, it occasionally produces two photons of frequency ωs and ωi. This process

is extremely inefficient and therefore produces a downconverted pair of photons for

∼ 107 or so pump photons. The key to obtaining efficient downconversion is to ensure

phase matching. Phase matching is related to momentum and energy conservation

laws applied to the the three-wave mixing process. The phase matching conditions

are given by,

ωp = ωs + ωi (7.4)

and

~kp = ~ks + ~ki (7.5)

where ~kp,s,i are momentum of the pump, signal and idler photons, given by, ~kp,s,i =

ωp,s,inp,s,ic

. In order to satisfy phase matching conditions, the crystal face needs to be

cut at an apposite angle with respect to the optic axis of the crystal called the phase

matching angle. The downconverted photons are correlated in terms of energy, linear

momentum and polarization degrees of freedom. SPDC can be classified in three types

viz. Type-0, Type I and Type-II. Type-0 SPDC entails the use of periodically poled

crystals. In the type-I process, the polarization state of the downconverted photons

are the same and it is orthogonal to the pump photon polarization. In type-II SPDC

process, these twin photons are orthogonally polarized, one of them have same po-

larization as that of the pump photon [105]. Though the downconverted photons are

entangled with respect to various internal degrees of freedom, for the present require-

ment, we focus on the polarization degree of freedom. Careful spatial and spectral

filtering is required to ensure that the signal and idler photons are indistinguishable

from each other.

111

Spontaneous parametric down-conversion process Chapter 7

Figure 7.1 – Two cemented BBO crystals are used to generate polarization entangledphotons, whose optics axes are oriented 90◦ with respect to each other. Two overlappingcones are produced, one in each crystal.

In the type-I SPDC, the downconverted photons are emitted as a cone. To produce

the polarized entangled photons, two negative uniaxial crystals are required. These

two identical crystals are cemented together by orienting their optical axis orthogonal

to each other. When the pump photon is vertically polarized, it is e-polarized for

the first crystal and o polarized with respect to the second. Care should be taken to

focus the pump waist at the interface between the two crystals. Thus, two horizontal

o-polarized photons are produced in the first crystal when the pump photon is verti-

cally polarized. Similarly, for horizontally polarized pump photons, the second crystal

produces two vertically polarized downconverted photons. With this configuration,

when the incident light is diagonally polarized, having both the e and o polarization

components, that is the pump photon is superposition of both the vertical and hori-

zontal components, b|H〉+ a|V 〉, the downconverted photons are entangled, and their

composite state can be written as,

|ψ〉 = a|HH〉+ eiφb|V V 〉 (7.6)

Here, the downconverted photons are produced in two cones as shown in Fig (7.1).

The photons should be collected from diametrically opposite regions of the overlapped

emission cones. If this is not done, the coincidence rates would be very poor even

when single photon counts in the individual SPADs are high. Great care must also

112


be taken to ensure walk-off compensation [106, 107]. The collection optics should

provide for filters and single mode fibers to ensure spectral and spatial filtering. The

latter step ensures the indistinguishability criterion which is necessary for a high-

quality entangled bi-photon source. The filters also ensure that the SPAD units are

not damaged due to excessive exposure to ambient light. The power supplies to our

SPADs also ensure this electronically.

Figure 7.2 – Experimental set up to generate a pair of polarization entangled photonsusing spontaneous parametric downconversion.

7.3 Experimental set up

Projector QWP Polarizer|H〉〈H| 0 0|V 〉〈V | 90 90|D〉〈D| 45 45|A〉〈A| −45 −45|R〉〈R| 45 90|L〉〈L| −45 90

Table 7.1 – Realizing different single qubit projection operators using quarter waveplate and polarizers.

In the experimental arrangement, a diode laser of wavelength 405 nm with a maxi-

mum output power of 100 mW is used as the pump. This vertically polarized pump

beam is collimated and suitable beam shaping optics is incorporated to ensure a

circular TEM00mode. A cemented pair of BBO (β − BaB2O4) crystals of 0.5 mm

113

Experimental set up Chapter 7

thickness each is used to produce polarization entangled photons. The crystal pair is

phase matched for degenerate downconversion, therefore the downconverted photons

are emitted with a wavelength of 810 nm. The downconverted photons emerge at a

half angle of 3◦ with respect to the pump beam. A half wave plate (HWP) is used

before the BBO crystal to rotate the polarization state of the pump beam. A walk-off

compensation plate made of quartz is placed between HWP and the BBO crystal to

adjust the relative phase between the horizontal and the vertical polarization compo-

nents of the pump beam [108]. A long pass filter is kept just after the BBO, to cut-off

all the lower wavelength components, transmits only photons of wavelength higher

than 780 nm. The downconverted photons are collected through a fiber coupler set-

up and the single mode fiber is connected to single photon avalanche photodetectors

(SPAD). Proper mode matching between the fiber and coupler is essential to ensure

high count rates. The fiber connectorized SPAD units typically have an efficiency

of ∼ 50% @810 nm with dark count rates less than 50 cps. Both the detectors are

connected to a FPGA based coincidence counting unit. The coincidence time window

is the maximal time interval between the arrival of signal and an idler photon pulses

and was fixed at 10ns [107]. The different Bell states can be produced using the half

wave plates and adjusting the pre-compensation plate. The next task was to carry

out projective measurements for the reconstruction of the state. The quantum state

of the two qubit systems is reconstructed through a QWP and polarizer pair placed in

each of the arms [109]. Since the present scheme of quantum state tomography does

not entail any joint two-photon measurement called a bell-state measurements, single

qubit projectors are realized in the individual arms by orienting the QWP and polar-

izer at appropriate angles. Table (7.1) provides a realization of different single qubit

projection operators. For complete quantum state reconstruction of the single qubit

systems, 3 complementary measurements are required and the measurements are σx,

σy and σz. Therefore, for two qubit systems, 9 measurements σi ⊗ σj are required

and each measurement consists of four projective measurements. Hence, the recon-

114


struction of the DWF of the two qubit case involves 36 projective measurements [49].

The details of the quantum state reconstruction are given in the following section.

7.4 Reconstruction of two-qubit discrete Wigner

functions

Figure 7.3 – A point α in the 4× 4 phase space φ can be identified by the set of twopoints α1 and α2 corresponding to φ1 and φ2 respectively. Therefore, every point α inphase space can be written as an ordered set of phase space points α = (α1, α2). Forexample, the point α = ((1, 0), (0, 1)) in φ is the point (0, 1) in the square (1, 0). Thisis shown as a dot in the figure.

In this section, we briefly discuss the direct reconstruction procedure for the discrete

Wigner function of the two-qubit systems [49]. It may be recalled from chapter-III,

that for the prime power dimensional case, N + 1 MUBs are guaranteed to exist

[84, 110, 111]. Therefore, the state of the system is completely reconstructed by

performing (N+1) MUB measurements. In the case of a two qubit-system, the discrete

phase space is a 4× 4 array of real elements and the dimension is N = 22, the DWF

can be reconstructed from 5 MUB measurements as discussed in Chapter-III [31].

Adamson et al., have given a method to perform these measurements using a Hong-Ou-

Mandel interferometer (HOMI) set up. In this measurement scheme, three of the five

measurements require only standard separable state measurements but the remaining

two involve two-photon Bell-state measurements using HOMI [50]. Alternatively, the

two qubit system can also be treated as having a composite dimension N = N1N2,

where Ni = 2. For each qubit, one can describe the phase space φi, whose points

115

Reconstruction of two-qubit discrete Wigner functions Chapter 7

are labelled by elements of the finite field F2 = {0, 1}. The phase space φ for the

composite system is defined as the Cartesian product of the phase space of individual

qubits, φ = φ1 ⊗ φ2. Therefore, every point α in phase space can be written as an

ordered set of phase space points α = (α1, α2), where the point αi corresponds to φi.

A point α in φ can be identified by the set of two points α1 and α2 corresponding

to φ1 and φ2 respectively. The first point α1 in this ordered set α = (α1, α2) belongs

to specific square in φ and α2 is point located within this square. For example, the

point α = ((1, 0), (0, 1)) in φ is the point (0, 1) in the square (1, 0), it shown in Fig

(7.3). Since, the concept of a line is not meaningful in composite case, the concept

of a slice is introduced. A slice λ in phase space is defined from the lines λi of the

phase space φi of individual qubits as, λ = (λ1, λ2). Therefore a slice λ is a set of

points α which contained in the lines λ1 and λ2 of φ1 and φ2. Let (α1, λsub) be a line

in the square α1. Let us now define a number p(α1, λsub), which is the sum of DWF

elements contained in the line λsub. This can be computed from the probabilities pλ,

which is the outcome of (N1 + 1)(N2 + 1) measurements. The number p(α1, λsub) can

be computed by summing all the probabilities pλ’s such that λ = (λ1, λsub) and λ1

contains β. In φ1 there are N1 + 1 such lines (λ1) passing through the point β, one in

each foliation. This results in sum of N1 + 1 such pλ’s. We can write this sum as,

∑λ=(λ1,λsub)|β∈λ1

pλ = (N1 + 1)p(β, λsub) +∑α1 6=β

p(α1, λsub) (7.7)

= N1p(β, λsub) +∑α1

p(α1, λsub) (7.8)

The last term in Eq (7.8) is sum of all the Wigner elements in the phase space φ1.

This can also be computed by adding the Wigner elements along any foliation F in

116


the phase space φ1. It is given by,

∑α1

p(α1, λsub) =∑

λ=(λ1,λsub)|λ1∈F

pλ (7.9)

By solving Eq (7.8) and Eq (7.9), simple expression can be given for p(β, λsub) as,

p(β, λsub) =1

N1

∑λ=(λ1,λsub)|β∈λ1

pλ +∑

λ=(λ1,λsub)|λ1∈F

pλ

(7.10)

Eq (7.10) helps us to calculate the sum of two DWF elements from the outcome of the

projective measurements pλ. From p(β, λsub), we can compute the individual DWF

elements. Using the same procedure the number p(β, λsub) can be computed for all

the lines in the square β. Therefore, the DWF at a point (β, µ) can be computed

from this number p(β, λsub) as,

W(β,µ) =1

N2

∑λsub)|µ∈λsub

p(β, λsub) +∑

λsub∈F

p(β, λsub)

(7.11)

So, using a recursive procedure one may compute the DWF at any point in phase

space using Eq (7.11). First, the sum of two DWF elements p(β, λsub) is computed

from the outcomes of the measurements pλ, which is a sum over four DWF elements.

Further, by calculating the p(β, λsub) for all the λsub in the square β in φ we have

given a expression to compute DWF at a point (β, µ).

117

Results and Discussion Chapter 7

7.5 Results and Discussion

Figure 7.4 – Experimentally reconstructed DWF of bi-photon polarization entangledstate.

Figure 7.5 – Real part of the experimentally reconstructed density matrix of bi-photonpolarization entangled state.

118


Figure 7.6 – Imaginary part of the experimentally reconstructed density matrix ofbi-photon polarization entangled state.

The DWF of the downconverted photons are completely described by performing 36

projective measurements on the two photon polarization entangled states. Table (7.2),

gives the single photon counts and the coincidence counts along this 36 projective

measurements. Table (7.3) gives the coincidence counts of the 9 measurements and

each measurement consisted of 4 projective measurements. Using these measurements

and the procedure given in the previous section, we have reconstructed the DWF of

the bi-photon polarization entangled states. DWF of the reconstructed state is given

in table (7.11) and its plot is given in Fig (7.4).

From this DWF, we have reconstructed the “density matrix” of the two qubit system,

using the inversion formula ρ =∑α

WαAα, it is given by,

ρ =

0.4847 0.0010− 0.0075i 0.0574− 0.0188i −0.4628 + 0.0471i

0.0010 + 0.0075i 0.0345 0.0020− 0.0036i −0.0120− 0.0262i

0.0574 + 0.0188i 0.0020 + 0.0036i −0.0057 −0.0679− 0.0050i

−0.4628− 0.0471i −0.0120 + 0.0262i −0.0679 + 0.0050i 0.4865

(7.12)

We may note here that the reconstructed“density matrix”has unit trace but one of the

119

Results and Discussion Chapter 7

eigenvalues is negative. The non-positivity arises from experimental errors. Further

processing through a maximum likelihood needs to be carried out on the data. Since

our purpose here is merely to demonstrate the theoretical procedure, we have not

carried out any further error analysis of the data. Rather, we show the computation

of the spin-flipped DWF, computation of the Stokes vector from the DWF and partial

trace procedure provided in the previous chapters.

PSingle 1

(cps)

Single 2

(cps)CC (cps)

HH 28986 29282 1535

HV 28874 37884 52

HD 29277 33898 904

HA 29231 33633 648

HR 28932 33638 930

HL 29198 339900 797

VH 38300 28746 40

VV 39252 38629 1565

VD 38913 34031 745

VA 38825 33776 934

VR 39347 33885 822

VL 39176 34104 673

DH 34591 28983 947

DV 34720 38433 740

DD 34841 33848 81

DA 35189 33953 1645

DR 35062 33914 1042

DL 35189 34201 770

P Single 1(cps)Single 2

(cps)CC (cps)

AH 32260 28512 638

AV 32570 38473 875

AD 32394 33505 1520

AA 32325 34967 48

AR 32313 33285 714

AL 32400 33591 777

RH 32371 28512 705

RV 32265 38713 801

RD 32395 33843 678

RA 32976 34588 953

RR 32592 33583 74

RL 32520 34070 1578

LH 32651 29410 805

LV 32433 39295 854

LD 31836 33720 785

LA 32564 34517 782

LR 32370 33918 1542

LL 32327 34229 40

Table 7.2 – Single and coincidence counts (CC) for 36 projective measurements.

HH 1535HD 904HR 930DH 947DD 81DR 1042RH 705RD 678RR 74

HV 52HA 648HL 797DV 740DA 1645DL 770RV 801RA 953RL 1578

VH 40VD 745VR 822AH 638AD 1520AR 714LH 805LD 785LR 1542

VV 1565VA 934VL 673AV 875AA 48AL 777LV 854LA 782LL 40

Table 7.3 – Coincidence counts associated with the 9 measurements, each measure-ment consisted of 4 projective measurements. For example, coincidence counts given inthe first line is associated with the σz ⊗ σz measurement.

120


AA -0.0111 -0.0083 0.0194 0.0303AD 0.2153 0.0185 -0.0415 0.2321DA 0.2566 0.0213 0.0142 0.2444DD 0.0238 0.0030 0.0023 -0.0202

DWF HH HV VH VV

Table 7.4 – Reconstructed DWF of the two photon polarization entangled states.

AA -0.0072 -0.0320 -0.0160 0.0215AD 0.2302 0.0043 -0.0249 0.2147DA 0.2453 -0.0116 0.0446 0.2582DD 0.0163 0.0097 -0.0093 -0.0078

DWF HH HV VH VV

Table 7.5 – DWF associated with the spin flipped system.

A 0.2144 0.2403D 0.3047 0.2407

WA H V

A 0.2666 0.3002D 0.2123 0.2210

WB H V

Table 7.6 – DWF of the sub-systems A and B computed from the reduction formula.

The plot for the real and imaginary part of the density matrix of the system is

given in Fig (7.5) and (7.6). Using the HWP and pre-compensation plate placed

before the BBO crystal we can produce the desired two photon entangled states. We

have oriented the HWP at 45◦ angle and tilted WP to produce the Bell state |ψ〉 =

1√2

(|HH〉 − |V V 〉). The closeness of the reconstructed state ρ with the desired state

|ψ〉 can be calculated from the quantity called fidelity, defined as F (ρ, |ψ〉) = 〈ψ|ρ|ψ〉

[51, 112]. Let W be the DWF of the reconstructed state and WB be the DWF of the

desired state. In the case of DWF, the fidelity of the reconstructed DWF W with

respect to the DWF WB of the desired state is given by, F (W,WB) =∑α

WαWBα . We

have found that the reconstructed state is very close to the Bell state with Fidelity

F (W,WB) = 0.9484. Mixedness of a quantum state can be computed from the linear

entropy of the state, it is defined as SL (ρ) =√

NN−1

[1− Tr (ρ2)] [61, 62]. In the

case of DWF it is defined as, SL (W ) =√

NN−1

[1−W TW ]. We have calculated the

linear entropy of the reconstructed DWF as SL (W ) = 0.3184. From the reconstructed

121


DWF, we have calculated the DWF associated with the spin flipped system, using

the Hadamard matrix constructed in Chapter-IV and it is given in Table (7.5). We

have computed the Stokes vector corresponding to the reconstructed DWF using the

Hadamard matrices defined in Chapter-V, and it is given by,

S = HW (7.13)

S =(1.0000 0.0907 0.0899 0.0381 − 0.1335 0.0249 − 0.0421 − 0.9217

− 0.1013 0.1389 − 0.0869 0.9295 − 0.0147 0.1377 0.0049 0.9423)T (7.14)

Using the reduction formula given in Chapter-VI, we have calculated the DWF of the

sub-systems A and B, it is given in Table (7.6).

7.6 Conclusions

All the analysis and quantities of interest presented in this chapter have been com-

puted purely in terms of the DWF which was obtained through the tomographic

reconstruction of the experimental data. This is in the spirit of treating DWF as a

valid representation of the state in its own right. The experimental results presented

in this chapter have illustrated all the theoretical findings presented earlier. There

are however some niggling problems with respect to the experimentally reconstructed

DWF with respect to experimental errors. In the standard tomographic reconstruc-

tion of the density matrix, there are sources of experimental errors and as a result, the

reconstructed matrix quite often does not have unit trace and is not positive either

as one or more of its eigenvalues turn out to be negative. A maximum likelihood

algorithm is implemented to reconstruct the best approximation of the experimental

data to a valid density matrix [109]. Such algorithms are based on demanding that

the reconstructed matrix should be positive and normalized. A similar optimization

122


algorithm is however not available for the DWF. We leave this issue for future work.

However, the central theoretical results obtained in chapters IV, V and VI have been

demonstrated for the experimental data. The second issue pertains to the optimality

of the experimental procedure of QST. It has been pointed out that state tomography

of density matrices based on complete MUB including joint measurements may be

optimal. This again is a matter for further experimental investigation.

123

Chapter 8

Summary, conclusions and futuredirections

8.1 Summary and conclusions

The rapid developments in areas of quantum information science and quantum com-

putation have contributed greatly to the deeper study of finite dimensional quantum

states. In the last couple of decades, investigations of discrete quantum systems has

proved to be a veritable treasure trove of results. These results could have radical and

far-reaching implications to the way we compute and transact information. Taking

cue from continuous systems where quantum mechanics was reformulated in terms of

a quantum phase space, several investigators have attempted to formulate phase-space

versions of discrete quantum states. The Wigner function in particular has proved to

be extremely useful in quantum optics. The Discrete Wigner Function construction

developed by Wootters and Gibbons et al., has been investigated in this thesis with

a view to investigating entanglement in multiqubit systems. The Wootters’ construc-

tion develops the DWF over a finite field and therefore has a rich algebraic structure

which in itself warrants deeper study.

Density matrices are routinely used in the study of entanglement detection, entangle-

ment characterization and in the study of decoherence purely in terms of DWFs. In

the present work, we have addressed some of these issues purely in terms of DWF.

The spin flip operation is an anti-unitary operation which is important in computing

125

Summary and conclusions Chapter 8

the concurrence of general two qubit systems. Also the overlap between the state

and its spin flipped state of the multi-qubit system is known as the n-concurrence or

Minkowskian squared norm, which is a good measure of entanglement and SLOCC

invariant. In chapter-IV, we addressed the method of performing spin flip operation

for multi-qubit DWFs. We have shown that, for a multi-qubit systems, the DWF

and DWF associated with the spin flipped systems are related through a Hadamard

matrix and which is independent of the quantum net used. This result would be useful

in quantification of entanglement in pure multi-qubit systems and also in the some

specific classes of mixed two qubit systems.

Since quantum systems can be represented by a density operator, DWF or Stokes

vector, finding the relationship between different representations of a state helps in

carrying over the advantages in one representation to the other. In chapter-V, we

presented a relationship between Stokes vector and DWF and also pointed out the

advantage in finding out this inter-relationship. We have given a direct relationship

between DWF and Stokes vector of the multi-qubit systems. They were shown to be

related through Hadamard matrices, which depend on the choice of the quantum net.

It was shown that for each quantum net, there exists a unique Hadamard matrixH and

these Hadamard matrices are different from the one used for the spin flip operation H.

The set SnH was then defined as the collection of all Hadamard matrices associated with

each quantum net. The Hadamard matrix H was not contained in the set SnH , however,

interestingly the product HH was always an element of the set SnH . It was also shown

that the product HH takes the DWF to the Stokes vector of the spin flipped state.

As a first application of this inter-relationship between Stokes vector and DWF, we

have given a method to calculate the n-concurrence or Minkowskian squared norm

purely in terms of the DWF for a multi-qubit systems. As a second application of this

inter-relationship, in chapter-VI, we have given a generalized reduction formula for

computing the DWF of the sub-systems from the DWF of the multi-qubit systems.

126


It is equivalent to the partial trace formula in the density matrix formalism. This

reduction formula is useful in entanglement characterization problems of multi-qubit

DWFs and in the study of decoherence. To give the experimental flavor to the DWF,

we set up an source to generate bi-photon polarization entangled states through the

SPDC process and reconstructed its DWF and demonstrated some of our theoretical

results.

8.2 Future directions

Despite the results reported in this thesis and elsewhere in the literature with regards

to entanglement quantification methods for the multi-qubit systems, there are some

loose ends in entanglement detection in the discrete phase space setting. Franco et al.,

have observed that, a two qubit system is entangled if either the DWF or the DWF

after partial transpose have negative entries, but formal proof of this is unavailable

till date [74]. Our present approach allow us to perform the equivalent of a partial

transposition operation, this problem is being taken up for further investigations.

As a next case, we would like to study the phenomenon of decoherence from the

discrete phase space point of view. In the present situation we have some mathematical

tools to compute entanglement and to identify sub-systems for DWFs, therefore these

results will be useful in this study. We also like to investigate the application of DWF

to quantum computation protocols.

On the experimental side, though we have successfully reconstructed the DWF of two

photon polarization states, it requires 9×4 = 36 projective measurements. But, most

of the quantum information protocols require the reconstruction of quantum states

with minimal number of measurements with high fidelity. Adamson et al., have given a

state reconstruction process using Hong-Ou-Mandel interferometer (HOMI), and this

process requires 5 MUB measurements [50]. State reconstruction using MUBs reduces

the complexity and redundancy in the quantum state estimation process. In future,

127

Future directions Chapter 8

we would like to reconstruct the DWF of the bi-photon polarization states using this

HOMI configuration. One of the problems that arise from experimental sources of

error is that the reconstructed “state” is not a genuine density matrix and one has to

take recourse to numerical techniques to identify the actual density matrix closest to

the measured one. The fact that density matrices have to be positive operators of

unit trace is used for this purpose. A similar treatment for reconstructed DWF needs

to be developed and is proposed as future work.

128

References

[1] D. Griffiths, Introduction to Quantum Mechanics (Cambridge University Press,2016), ISBN 9781107179868.

[2] B. Claude Cohen-Tannoudji, Quantum Mechanics Volume 1 (Hermann, ????),ISBN 9782705683924.

[3] G. Martynov, Classical Statistical Mechanics, Fundamental Theories of Physics(Springer Netherlands, 2012), ISBN 9789401589635.

[4] W. Schleich, Quantum Optics in Phase Space (Wiley, 2015), ISBN9783527802555.

[5] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).

[6] A. M. O. de Almeida, Journal of Physics A: Mathematical and General 36, 67(2003).

[7] E. Calzetta, A. Roura, and E. Verdaguer, Physica A: Statistical Mechanics andits Applications 319, 188 (2003), ISSN 0378-4371.

[8] C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space: AnOverview with Selected Papers, World Scientific Series in 20th Century Physics(World Scientific, 2005), ISBN 9789814485876.

[9] E. Wigner, Phys. Rev. 40, 749 (1932).

[10] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).

[11] R. J. Glauber, Phys. Rev. 131, 2766 (1963).

[12] K. HUSIMI, Proceedings of the Physico-Mathematical Society of Japan. 3rdSeries 22, 264 (1940).

[13] R. Hudson, Reports on Mathematical Physics 6, 249 (1974), ISSN 0034-4877.

[14] A. Zavatta, S. Viciani, and M. Bellini, Science 306, 660 (2004).

[15] A. I. Lvovsky, Squeezed Light (John Wiley & Sons, Inc., 2015), pp. 121–163,ISBN 9781119009719.

[16] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett.98, 030502 (2007).

[17] R. Schnabel, Physics Reports 684, 1 (2017), ISSN 0370-1573, squeezed statesof light and their applications in laser interferometers.

129

[18] A. Kenfack and K. Zyczkowski, Journal of Optics B 6, 396 (2004).

[19] T. V. Gevorgyan, A. R. Shahinyan, and G. Y. Kryuchkyan, Phys. Rev. A 79,053828 (2009).

[20] G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).

[21] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70,1244 (1993).

[22] A. Royer, Foundations of Physics 19, 3 (1989), ISSN 1572-9516.

[23] U. Leonhardt, Measuring the quantum state of light (Cambridge UniversityPress, Cambridge, UK New York, 1997), ISBN 0521497302.

[24] A. Muller, H. Zbinden, and N. Gisin, EPL (Europhysics Letters) 33, 335 (1996).

[25] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Woot-ters, Phys. Rev. Lett. 70, 1895 (1993).

[26] L. K. Grover (1996), quant-ph/9605043.

[27] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).

[28] B. Schumacher, Phys. Rev. A 51, 2738 (1995).

[29] L. Cohen and M. Scully, Foundations of Physics 16, 295 (1986), ISSN 0015-9018.

[30] D. Galetti and A. T. Piza, Physica A: Statistical Mechanics and its Applications186, 513 (1992), ISSN 0378-4371.

[31] K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101(2004).

[32] W. Wootters, Foundations of Physics 16, 391 (1986), ISSN 0015-9018.

[33] W. K. Wootters, Annals of Physics 176, 1 (1987), ISSN 0003-4916.

[34] W. K. Wootters, IBM J. Res. Dev. 48, 99 (2004), ISSN 0018-8646.

[35] R. P. Feynman, in Quantum Implications: Essays in Honour of David Bohm,edited by B. J. Hiley and D. Peat (Methuen, 1987), pp. 235–248.

[36] S. Chaturvedi, E. Ercolessi, G. Marmo, G. Morandi, N. Mukunda, and R. Simon,Pramana 65, 981 (2005).

[37] O. Cohendet, P. Combe, M. Sirugue, and M. Sirugue-Collin, Journal of PhysicsA: Mathematical and General 21, 2875 (1988).

[38] P. Bianucci, C. Miquel, J. Paz, and M. Saraceno, Physics Letters A 297, 353(2002), ISSN 0375-9601.

[39] J. P. Paz, A. J. Roncaglia, and M. Saraceno, Phys. Rev. A 72, 012309 (2005).

[40] C. Cormick, E. F. Galvao, D. Gottesman, J. P. Paz, and A. O. Pittenger, Phys.Rev. A 73, 012301 (2006).

130

quant-ph/9605043

[41] J. P. Paz, Phys. Rev. A 65, 062311 (2002).

[42] M. Koniorczyk, V. Buzek, and J. Janszky, Phys. Rev. A 64, 034301 (2001).

[43] C. C. Lopez and J. P. Paz, Phys. Rev. A 68, 052305 (2003).

[44] R. W. Spekkens, Phys. Rev. A 75, 032110 (2007).

[45] E. F. Galvao, Phys. Rev. A 71, 042302 (2005).

[46] V. Veitch, C. Ferrie, D. Gross, and J. Emerson, New Journal of Physics 14,113011 (2012).

[47] V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New Journal ofPhysics 16, 013009 (2014).

[48] I. D. Ivonovic, Journal of Physics A: Mathematical and General 14, 3241 (1981).

[49] R. Asplund and G. Bjork, Phys. Rev. A 64, 012106 (2001).

[50] R. B. A. Adamson and A. M. Steinberg, Phys. Rev. Lett. 105, 030406 (2010).

[51] M. Nielsen and I. Chuang, Quantum Computation and Quantum Informa-tion: 10th Anniversary Edition (Cambridge University Press, 2010), ISBN9781139495486.

[52] M. Holmes and W. Schudy, Thesis:Bachelor of science, Worcester PolytechnicInstitute (2005).

[53] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009).

[54] R. F. Werner, Phys. Rev. A 40, 4277 (1989).

[55] A. Peres, Phys. Rev. Lett. 77, 1413 (1996).

[56] M. Horodecki, P. Horodecki, and R. Horodecki, Physics Letters A 223, 1 (1996),ISSN 0375-9601.

[57] S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997).

[58] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information,Cambridge Series on Information and the Natural Sciences (Cambridge Univer-sity Press, 2000), ISBN 9780521635035.

[59] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev.A 54, 3824 (1996).

[60] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

[61] E. Santos and M. Ferrero, Phys. Rev. A 62, 024101 (2000).

[62] S. Bose and V. Vedral, Phys. Rev. A 61, 040101 (2000).

[63] W. B. Case, American Journal of Physics 76, 937 (2008).

[64] J. Schwinger, Proceedings of the National Academy of Sciences 46, 570 (1960).

131

[65] F. A. Buot, Phys. Rev. B 10, 3700 (1974).

[66] J. Hannay and M. Berry, Physica D: Nonlinear Phenomena 1, 267 (1980), ISSN0167-2789.

[67] J. A. Vaccaro and D. T. Pegg, Phys. Rev. A 41, 5156 (1990).

[68] U. Leonhardt, Phys. Rev. Lett. 74, 4101 (1995).

[69] U. Leonhardt, Phys. Rev. A 53, 2998 (1996).

[70] A. Luis and J. Perina, Journal of Physics A: Mathematical and General 31,1423 (1998).

[71] C. Miquel, J. P. Paz, and M. Saraceno, Phys. Rev. A 65, 062309 (2002).

[72] A. Casaccino, E. F. Galvao, and S. Severini, Phys. Rev. A 78, 022310 (2008).

[73] A. O. Pittenger and M. H. Rubin, Journal of Physics A: Mathematical andGeneral 38, 6005 (2005).

[74] R. Franco and V. Penna, Journal of Physics A: Mathematical and General 39,5907 (2006).

[75] J. Ryu, J. Lim, S. Hong, and J. Lee, Phys. Rev. A 88, 052123 (2013).

[76] G. Bjork, J. L. Romero, A. B. Klimov, and L. L. Sanchez-Soto, J. Opt. Soc.Am. B 24, 371 (2007).

[77] A. Klimov, J. Romero, G. BjA¶rk, and L. SA¡nchez-Soto, Annals of Physics324, 53 (2009), ISSN 0003-4916.

[78] A. B. Klimov, C. Munoz, and J. L. Romero, Journal of Physics A: Mathematicaland General 39, 14471 (2006).

[79] D. Gross, Journal of Mathematical Physics 47, 122107 (2006).

[80] C. Cormick and J. P. Paz, Phys. Rev. A 74, 062315 (2006).

[81] C. MUNOZ, A. B. KLIMOV, L. L. SANCHEZ-SOTO, and G. BJORK, Inter-national Journal of Quantum Information 07, 17 (2009).

[82] A. Mari and J. Eisert, Phys. Rev. Lett. 109, 230503 (2012).

[83] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications(Cambridge University Press, 1994), ISBN 9780521460941.

[84] Bandyopadhyay, Boykin, Roychowdhury, and Vatan, Algorithmica 34, 512(2002), ISSN 1432-0541.

[85] N. Gisin and S. Popescu, Phys. Rev. Lett. 83, 432 (1999).

[86] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).

[87] G. Jaeger, M. Teodorescu-Frumosu, A. Sergienko, B. E. A. Saleh, and M. C.Teich, Phys. Rev. A 67, 032307 (2003).

132

[88] G. Jaeger, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 68,022318 (2003).

[89] S. S. Bullock and G. K. Brennen, Journal of Mathematical Physics 45, 2447(2004).

[90] V. Buzek, M. Hillery, and R. F. Werner, Phys. Rev. A 60, R2626 (1999).

[91] F. De Martini, V. Buzek, F. Sciarrino, and C. Sias, Nature 419, 815 (2002),ISSN 0028-0836.

[92] K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58,883 (1998).

[93] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).

[94] D. A. Meyer and N. R. Wallach, Journal of Mathematical Physics 43, 4273(2002).

[95] A. Wong and N. Christensen, Phys. Rev. A 63, 044301 (2001).

[96] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).

[97] P. Rungta, V. Buzek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev.A 64, 042315 (2001).

[98] B. M. Terhal, Physics Letters A 271, 319 (2000), ISSN 0375-9601.

[99] H. G. Berry, G. Gabrielse, and A. E. Livingston, Appl. Opt. 16, 3200 (1977).

[100] E. Collett, Optics Communications 52, 77 (1984), ISSN 0030-4018.

[101] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64,052312 (2001).

[102] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A53, 2046 (1996).

[103] R. Boyd, Nonlinear Optics, Nonlinear Optics Series (Elsevier Science, 2008),ISBN 9780080485966.

[104] D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970).

[105] D. Dehlinger and M. W. Mitchell, American Journal of Physics 70, 898 (2002).

[106] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys.Rev. A 60, R773 (1999).

[107] R. Rangarajan, PhD Thesis, University of Illinois at Urbana (2010).

[108] P. Trojek, PhD Thesis, Ludwig-Maximilian University of Munich (2007).

[109] J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Advances in Atomic Molecularand Optical Physics 52, 105 (2005).

[110] W. K. Wootters and B. D. Fields, Annals of Physics 191, 363 (1989), ISSN0003-4916.

133

[111] J. Lawrence, i. c. v. Brukner, and A. Zeilinger, Phys. Rev. A 65, 032320 (2002).

[112] C. A. Fuchs and J. van de Graaf, IEEE Transactions on Information Theory45, 1216 (1999), ISSN 0018-9448.

134

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