Entanglement Measures in Quantum Computing

About distinguishable and indistinguishable particles, entanglement, exchange and

correlation

Szilvia NagyDepartment of Telecommunications,Széchenyi István University, Győr

Péter Lévay, János Pipek, Péter Vrana, Szilárd Szalay,

Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest

Contents

• Motivation• Realization of entangled states• Distinguishable and indistinguishable particles

propertiesentanglement’s two facemeasures for entanglement

Schmidt and Slater ranksConcurrence and Slater correlation

measureentropies

• Generalization, entanglement types in three or more particle systems

Motivation

• Entanglement plays an essential role in paradoxes and counter-intuitive consequences of quantum mechanics.

• Characterization of entanglement is one of the fundamental open problems of quantum mechanics.

• Related to characterization and classification of positive maps on C* algebras.

• Applications of quantum mechanics, like quantum computingquantum cryptographyquantum teleportation

is based on entanglement. “Entanglement lies in the heart of quantum computing.”

Physical systems

• Quantum dots: the charge carriers are confined /restricted/ in all three

dimensionsit is possible to control the

number of electrons in the dotsthe qubits can be of orbital or

spin degrees of freedomtwo qubit gates can be e.g. magnetic field

• Neutral atoms in magnetic or optical microtraps

Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)

Distinguishable and indistinguishable particles

• Not identical particles• Large distance or

energy barrier• No exchange effects

arise

• Identical particles• Small distance and

barrier• Exchange properties

are essential

Distinguishable particles

Small overlap between and The exchange contributions are small in the

Slater determinants

BAABi

For two particles and two states A B

Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)

Indistinguishable particles

Large overlap between and The exchange contributions are significant in the

Slater determinants

21211

2

1 t

If the energy barrier is lowered

Indistinguishable particles

A mixed state of two Slater determinants arises

Suppose, that after time evaluation

2121

21212

2

1

t

Distinguishable particles

We get one of the Bell states

Rising the barrier again – increasing the distance

BABAABi

2

1

What is entanglement?

Basic concept: two subsystems are not entangled if and only if both constituents possess a complete set of properties.→separability of wave functions in Hilbert space

Distinguishable particlesthe two subsets are not entangled, iff the system’s Schmidt rank r is 1, i.e. only one non-zero coefficient is in the Schmidt decomposition.

Indistinguishable particles the two subsets are not entangled, iff the system’s Slater rank is 1, i.e. only one non-zero coefficient is in the Slater decomposition.

1 ,0 2

1

i

ii

r

iiii zzbaz

1Tr2 , 0 T

PPPPccP

Distinguishable and indistinguishable particles

• Not identical particles• Large distance or

energy barrier• No exchange effects

arise• Schmidt

decomposition

→Schmidt rank

• Identical particles• Small distance and

barrier• Exchange properties

are essential• Slater decomposition

→Slater rank

i

iii baz

i

ccP 0

Distinguishable particles - concurrence

The state can be written as

The concurrence is

Concurrence can also be introduced for indistinguishable particles.

Magic basis for two particles

2322

21

121

0

imim

mm

mj

jj

2j

jC

Indistinguishable particles – η measure

Both C and are 0 if the states are not entangled and 1 if maximally entangled.

The definition of the Slater correlation measure

0 ccP

PP

if

Schliemann & al. Phys. Rev. A 64 022303 (2001)

Distinguishable and indistinguishable particles

• Not identical particles• Large distance or

energy barrier• No exchange effects

arise• Schmidt

decomposition

→Schmidt rank• concurrence

• Identical particles• Small distance and

barrier• Exchange properties

are essential• Slater decomposition

→Slater rank measure

i

iii baz

i

ccP 0

Von Neumann and Rényi entropies

In our case

Good correlation measures for fermions. The von Neumann entropy is

And the th Rényi entropies are

0 Trlog1

1

logTr

2

21

ρS

ρρS

2

21

2

221

11 with

0 1log1

11

1log1log1

x

xxS

xxxxS

The minimum of the entropy

According to Jensen’s inequality

thus the von Neumann entropy is

and S=1 iff =0, i.e., if the Slater rank is 1.

MSN

MSN

212

222

loglog

loglog

NS

SS

MS

N 21

22

22

12

2

logloglog

log0

It can be shown that

Distinguishable and indistinguishable particles -

summary• Not identical particles• Large distance or

energy barrier• No exchange effects

arise• Schmidt decomposition

→Schmidt rank• Concurrence

• Smin=0

• Identical particles• Small distance and

barrier• Exchange properties

are essential• Slater decomposition

→Slater rank• η measure

• Smin=1

i

iii baz

i

ccP 0

The measures of entanglementThe connection between the entropy and the concurrence for specially

parameterized two-electron states:

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

The measures of entanglementThe connection between the concurrence and for

specially parameterized two-electron states:

221 12C

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

The measures of entanglement

The connection between the entropy and for specially parameterized two-electron states:

2

21

221

11 with

1log1log1

x

xxxxS

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

Three fermions

With

There are at least two essentially different types of entanglement if three or more particles are present.

3 particles, 6 one-electron states

ijk

kliijlkiij

P

jlkPBPA

PP

DetBDetABTrATrABT

~

,),(),(

,

444

321123

##2123

And the “dual state”

Lévay& al. Phys. Rev. A 78, 022329 (2008)

3 particles, 6 one-electron states:

Non-entangled states (separable or biseparable):

Entangled state type 1

Entangle state type 2

Three fermions

321

123 .0~,0

eeeP

PT ijk

213132321321

123

21

.0

eeeeeeeeeeeeP

T

Lévay& al. Phys. Rev. A 78, 022329 (2008)

• Developing a series of measures useable for any particles with any (finite) one-fermion states

• Basis: Corr by Gottlieb&Mauser

• Generalization: the distance not only from the uncorrelated statistical density matrix, but from characteristic correlated ones.

Future plans

A.D. Gottlieb& al. Phys. Rev. Lett 95, 123003 (2005)

Recent publications by the group

•Lévay, P., Nagy, Sz. and Pipek, J.,Elementary Formula for Entanglement Entropies of Fermionic Systems,Phys. Rev. A, 72, 022302 (2005).

•Szalay, Sz., Lévay, P., Nagy, Sz., Pipek, J.,A study of two-qubit density matrices with fermionic purifications,J. Phys. A - Math. Theo., 41, 505304, (2008).

•Lévay, P., Vrana, P.,Three fermions with six single particle states can be entangled in two inequivalent ways,Phys.Rev. A, 78 022329, (2008).