Postulates of Postulates of Quantum Quantum MechanicsMechanics

SOURCESSOURCESAngela Antoniu, David Fortin, Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Artur Ekert, Michael Frank, Kevin Irwig , Kevin Irwig , Anuj Dawar , Anuj Dawar , Michael NielsenMichael NielsenJacob Biamonte and studentsJacob Biamonte and students

Linear OperatorsLinear Operators• VV,,WW: Vector spaces.: Vector spaces.

• A A linear operatorlinear operator AA from from VV to to WW is a linear function is a linear function AA::VVWW. An . An operator operator onon VV is an operator from is an operator from VV to itself. to itself.

• Given bases for Given bases for VV and and WW, we can represent linear operators as , we can represent linear operators as matricesmatrices..

• An operator An operator AA on on VV is is HermitianHermitian iff iff it is self-adjoint ( it is self-adjoint (AA==AA††). ). • Its Its diagonal elementsdiagonal elements are are realreal..

Short reviewShort review

Eigenvalues & EigenvectorsEigenvalues & Eigenvectors

• vv is called an is called an eigenvectoreigenvector of linear operator of linear operator AA iff iff AA just just multiplies multiplies vv by a scalar by a scalar xx, , i.e.i.e. AvAv=x=xvv – ““eigen” (German) = “characteristic”.eigen” (German) = “characteristic”.

• xx, the , the eigenvalueeigenvalue corresponding to eigenvector corresponding to eigenvector vv, is just , is just the scalar that the scalar that AA multiplies multiplies vv by. by.

• the eigenvaluethe eigenvalue x x is is degeneratedegenerate if it is shared by 2 if it is shared by 2 eigenvectors that are not scalar multiples of each other.eigenvectors that are not scalar multiples of each other.– (Two different eigenvectors (Two different eigenvectors have the same eigenvaluehave the same eigenvalue))

• Any Hermitian operator has all Any Hermitian operator has all real-valued eigenvectorsreal-valued eigenvectors, , which are which are orthogonalorthogonal (for distinct eigenvalues). (for distinct eigenvalues).

Exam ProblemsExam Problems

• Find eigenvalues and eigenvectors of operators.Find eigenvalues and eigenvectors of operators.• Calculate solutions for quantum arrays.Calculate solutions for quantum arrays.• Prove that rows and columns are orthonormal.Prove that rows and columns are orthonormal.• Prove probability preservationProve probability preservation• Prove unitarity of matrices.Prove unitarity of matrices.• Postulates of Quantum Mechanics. Examples and Postulates of Quantum Mechanics. Examples and

interpretations.interpretations.• Properties of unitary operatorsProperties of unitary operators

Unitary TransformationsUnitary Transformations• A matrix (or linear operator) A matrix (or linear operator) UU is is unitaryunitary iff its inverse iff its inverse

equals its adjoint: equals its adjoint: UU11 = = UU††

• Some properties of Some properties of unitary transformations (unitary transformations (UTUT))::– Invertible, bijective, one-to-one.Invertible, bijective, one-to-one.– The set of row vectors is The set of row vectors is orthonormal.orthonormal.– The set of column vectors is The set of column vectors is orthonormal.orthonormal.– Unitary transformation Unitary transformation preserves vector lengthpreserves vector length: :

||UU| = || = | ||• Therefore also Therefore also preserves total probabilitypreserves total probability over all states: over all states:

– UT corresponds to a UT corresponds to a change of basischange of basis, from one orthonormal basis , from one orthonormal basis to another.to another.

– Or, a Or, a generalized rotation ofgeneralized rotation of in Hilbert space in Hilbert space

i

is22

)(

Who an when invented all this stuff??

A great A great breakthroughbreakthrough

Postulates of Quantum MechanicsPostulates of Quantum MechanicsLecture objectivesLecture objectives

• Why are postulates important?Why are postulates important?– … … they provide the they provide the connectionsconnections between the physical, between the physical,

real, world real, world andand the the quantum mechanics mathematics quantum mechanics mathematics used to model these systemsused to model these systems

• Lecture ObjectivesLecture Objectives– Description of Description of connectionsconnections– Introduce the Introduce the postulatespostulates– Learn Learn how to usehow to use them them– ……and and when to usewhen to use them them

Physical SystemsPhysical Systems -- Quantum MechanicsQuantum Mechanics

ConnectionsConnections

Tensor product Tensor product of componentsof components

Composite physical Composite physical systemsystem

Postulate 4Postulate 4

Measurement Measurement operatorsoperators

Measurements of a Measurements of a physical systemphysical system

Postulate 3Postulate 3

Unitary Unitary transformationtransformation

Evolution of a physical Evolution of a physical systemsystem

Postulate 2Postulate 2

Hilbert SpaceHilbert Space

Isolated physical Isolated physical systemsystem

Postulate 1Postulate 1

Postulate 1:Postulate 1: State SpaceState Space

Systems and SubsystemsSystems and Subsystems• Intuitively speaking, a Intuitively speaking, a physical systemphysical system consists of a consists of a

region of spacetimeregion of spacetime && all the entities ( all the entities (e.g.e.g. particles & particles & fields) fields) contained within itcontained within it..– The The universeuniverse (over all time) is a physical system (over all time) is a physical system– Transistors, computers, peopleTransistors, computers, people: also physical systems.: also physical systems.

• One physical system A is a One physical system A is a subsystemsubsystem of another of another system B (write Asystem B (write AB) iff A is completely contained B) iff A is completely contained within B.within B.

• Later, we may try to make these definitions Later, we may try to make these definitions more more formal & preciseformal & precise..

A B

Closed vs. Open SystemsClosed vs. Open Systems• A subsystem isA subsystem is closedclosed to the extent that no particles, to the extent that no particles,

information, energy, or entropy information, energy, or entropy enter or leave the enter or leave the system.system.– The The universeuniverse is (presumably) a closed system. is (presumably) a closed system.– Subsystems Subsystems of the universe may be of the universe may be almostalmost closed closed

• Often in physics we consider statements about Often in physics we consider statements about closed closed systems.systems.– These statements may often be These statements may often be perfectlyperfectly true true only in a only in a

perfectlyperfectly closed closed system. system.– However, they will often also be However, they will often also be approximatelyapproximately true true in any in any

nearlynearly closed closed system (in a well-defined way) system (in a well-defined way)

Concrete vs. Abstract SystemsConcrete vs. Abstract Systems• Usually, when reasoning about or interacting with a system, an entity Usually, when reasoning about or interacting with a system, an entity

((e.g.e.g. a physicist) has in mind a a physicist) has in mind a descriptiondescription of the system of the system..

• A description that contains A description that contains everyevery property of the system is an property of the system is an exactexact or or concreteconcrete description.description.– That system (to the entity) is a That system (to the entity) is a concreteconcrete system. system.

• Other descriptions are Other descriptions are abstractabstract descriptions.descriptions.– The system (as considered by that entity) is an The system (as considered by that entity) is an abstractabstract system system, to some , to some

degree.degree.

• We We nearly alwaysnearly always deal with deal with abstract systemsabstract systems!!– Based on the descriptions that are available to us.Based on the descriptions that are available to us.

States & State SpacesStates & State Spaces• A A possible statepossible state SS of an abstract system of an abstract system AA (described by a (described by a

description description DD) is any concrete system ) is any concrete system CC that is consistent that is consistent with with DD..– I.e.I.e., it is possible that the system in question could be completely , it is possible that the system in question could be completely

described by the description of described by the description of CC..

• The The state spacestate space of of AA is the set of all possible states of is the set of all possible states of AA..

• Most of the class, the concepts we’ve discussed can be Most of the class, the concepts we’ve discussed can be applied to applied to eithereither classicalclassical or or quantumquantum physics physics– Now, let’s get to the Now, let’s get to the uniquely quantumuniquely quantum stuff… stuff…

An example of a state space

Schroedinger’s Cat and Schroedinger’s Cat and Explanation of QubitsExplanation of Qubits

Postulate 1 in a Postulate 1 in a simple way:simple way: An An isolated physical isolated physical system is described system is described by a by a unit vectorunit vector ( (state state vectorvector) in a Hilbert ) in a Hilbert space (space (state spacestate space))

Cat is isolated in the box

Distinguishability of StatesDistinguishability of States• ClassicalClassical andand quantum mechanicsquantum mechanics differdiffer regarding the regarding the

distinguishabilitydistinguishability of states of states..

• In In classical mechanicsclassical mechanics, there is no issue:, there is no issue:– Any two states Any two states ss, , tt are either the same ( are either the same (s s = = tt), or different (), or different (s s tt), ),

and that’s all there is to it. and that’s all there is to it.

• In In quantum mechanicsquantum mechanics ( (i.e.i.e. in realityin reality):):– There are pairs of states There are pairs of states s s tt that are that are mathematically distinctmathematically distinct,, but but

not 100% not 100% physically distinguishablephysically distinguishable..– Such states Such states cannot be reliably distinguishedcannot be reliably distinguished by by any number of any number of

measurements, no matter how precise.measurements, no matter how precise.• But you But you cancan know know the real state (with high probability), the real state (with high probability), ifif you prepared you prepared the the

system to be in a certain state.system to be in a certain state.

Postulate 1:Postulate 1: State Space State Space– Postulate 1 Postulate 1 defines “the setting” in which Quantum Mechanics takes defines “the setting” in which Quantum Mechanics takes

place.place.– This setting is the This setting is the Hilbert space.Hilbert space.– The Hilbert Space is anThe Hilbert Space is an inner product space which satisfies inner product space which satisfies the the

condition of completeness (condition of completeness (recall math lecture few weeks agorecall math lecture few weeks ago).).

• Postulate1:Postulate1: Any Any isolatedisolated physical spacephysical space is associated with a is associated with a complex vector spacecomplex vector space with inner productwith inner product called the called the State State SpaceSpace of the system. of the system. – The system is The system is completely describedcompletely described by a by a state vectorstate vector, a , a unit vectorunit vector, ,

pertaining to the state space. pertaining to the state space. – The state space describes all possible states the system can be in. The state space describes all possible states the system can be in. – Postulate 1 Postulate 1 does NOT telldoes NOT tell us either what us either what the state spacethe state space is or what is or what

the the state vectorstate vector is. is.

Distinguishability of States, more Distinguishability of States, more preciselyprecisely

• Two state vectors Two state vectors ss and and tt are are (perfectly) distinguishable(perfectly) distinguishable or or orthogonalorthogonal (write (write sstt) ) iffiff ss††tt = 0. = 0. (Their inner product is zero.) (Their inner product is zero.)

• State vectors State vectors ss and and tt are are perfectly indistinguishableperfectly indistinguishable or or identicalidentical (write (write ss==tt) )

iff iff ss††tt = 1 = 1. (Their inner product is one.). (Their inner product is one.)

• Otherwise, Otherwise, ss and and tt are both are both non-orthogonalnon-orthogonal, and , and non-identicalnon-identical. . Not Not perfectly distinguishableperfectly distinguishable..

• We say, “the We say, “the amplitudeamplitude of state of state ss, given state , given state tt, is , is ss††tt”. ”. – Note:Note: amplitudes are amplitudes are complex numberscomplex numbers..

s

t

State Vectors & Hilbert SpaceState Vectors & Hilbert Space• Let Let SS be any be any maximal setmaximal set of distinguishable possible of distinguishable possible

states states ss, , tt, …, … of an abstract system of an abstract system AA..

• Identify the elements of Identify the elements of SS with with unit-lengthunit-length, , mutually-mutually-orthogonalorthogonal (basis) vectors in an abstract complex vector (basis) vectors in an abstract complex vector space space HH..– The “Hilbert space”The “Hilbert space”

• Postulate 1:Postulate 1: The The possible statespossible states of of AAcan be can be identifiedidentified with the with the unitunitvectorsvectors of of HH. . s

t

Postulate 2:Postulate 2: EvolutionEvolution

Postulate 2:Postulate 2: Evolution Evolution• Evolution of an isolated systemEvolution of an isolated system can be expressed as: can be expressed as:

where where tt11, t, t22 are are moments in timemoments in time and and U(tU(t11, t, t22)) is a unitary is a unitary operator.operator.– UU may vary with time. Hence, the corresponding segment of time is may vary with time. Hence, the corresponding segment of time is

explicitly specified:explicitly specified:

U(tU(t11, t, t22))– the process is in a sense the process is in a sense MarkovianMarkovian ( (history doesn’t matterhistory doesn’t matter) and ) and

reversiblereversible, since , since

)v(t ) t, U(t )v(t 1212

v v UU† Unitary operations preserve inner product

Example of evolutionExample of evolution

Time EvolutionTime Evolution• Recall the Postulate:Recall the Postulate: (Closed) systems evolve (change state) over (Closed) systems evolve (change state) over

time via unitary transformations.time via unitary transformations.t2t2 = U = Ut1t1t2 t2 t1t1

• Note that since Note that since UU is is linearlinear, a , a small-factor changesmall-factor change in amplitude of a in amplitude of a particular state at particular state at t1t1 leads to a leads to a correspondingly small changecorrespondingly small change in the in the amplitude of the corresponding state at amplitude of the corresponding state at t2t2..

– ChaosChaos (sensitivity to initial conditions) requires an (sensitivity to initial conditions) requires an ensemble of initial statesensemble of initial states that that are different enough to be distinguishable (in the sense we defined)are different enough to be distinguishable (in the sense we defined)

• Indistinguishable initial states Indistinguishable initial states never beget distinguishable outcomenever beget distinguishable outcome

WavefunctionsWavefunctions• Given any set Given any set SS of system states (mutually of system states (mutually

distinguishable, or not), distinguishable, or not),

• A quantum state vector can also be translated to a A quantum state vector can also be translated to a wavefunctionwavefunction : : S S CC,, giving, for each state giving, for each state ssSS, , the the amplitude amplitude ((ss)) of that state. of that state.

is called a is called a wavefunctionwavefunction because its time evolution obeys an because its time evolution obeys an equation (equation (Schrödinger’s equationSchrödinger’s equation) which has the form of a ) which has the form of a wave equationwave equation when when SS ranges over a space of ranges over a space of positionalpositional states. states.

Schrödinger’s Wave Equation for particlesSchrödinger’s Wave Equation for particles

We have a system with states given byWe have a system with states given by ((xx,t,t)) where: where:– tt is a is a global timeglobal time coordinate, and coordinate, and– xx describes describes NN/3 particles/3 particles (p (p11,…,p,…,pNN/3/3) ) with masseswith masses ( (mm11,…,,…,mmNN/3/3) )

in a in a 3-D Euclidean3-D Euclidean space, space, – where each where each ppii is located at is located at coordinatescoordinates ( (xx33ii, x, x33ii+1+1, , xx33ii+2+2), and ), and – where particles interact with where particles interact with potential energy function potential energy function VV((xx,,tt),),

• the wavefunction the wavefunction ((xx,,tt) obeys the following (2) obeys the following (2ndnd-order, -order, linear, partial) differential equation:linear, partial) differential equation:

),(),(),(1

2

1

02

2

3/

tt

tVtxm

N

j jj

xxx

i

Planck Constant

Features of the wave equationFeatures of the wave equation

• Particles’ Particles’ momentummomentum state state pp is encoded implicitly is encoded implicitly by the by the particle’s wavelengthparticle’s wavelength :: pp==hh//

• The The energyenergy of any state is given by the of any state is given by the frequency frequency of rotation of rotation of the wavefunction in the complex of the wavefunction in the complex plane: plane: EE==hh..

• By By simulating this simple equationsimulating this simple equation, one can , one can observe basic quantum phenomena such as:observe basic quantum phenomena such as:– InterferenceInterference fringes fringes– TunnelingTunneling of wave packets through potential barriers of wave packets through potential barriers

Heisenberg and Schroedinger Heisenberg and Schroedinger views of Postulate 2views of Postulate 2

..in this class we are interested in Heisenberg’s view…..

This is Heisenberg picture

This is Schroedinger picture

The Solution to the Schrödinger EquationThe Solution to the Schrödinger Equation• The Schrödinger Equation governs the transformation of an The Schrödinger Equation governs the transformation of an

initial initial input stateinput state to a final to a final output state output state . . • It is a prescription for what we want to do to the computer. It is a prescription for what we want to do to the computer.

• is a time-dependent Hermitian matrix of size 2is a time-dependent Hermitian matrix of size 2n n called called the the HamiltonianHamiltonian

• is a matrix of size 2is a matrix of size 2nn called the called the evolution matrixevolution matrix, ,

• Vectors of complex numbers of length 2Vectors of complex numbers of length 2nn

• TTττ is the time-ordering operatoris the time-ordering operator

0ˆ0ˆexp0

tUdHiTt

t

H

tU

0 t

The Schrödinger EquationThe Schrödinger Equation

• nn is the number of quantum bits (qubits) in the quantum computer is the number of quantum bits (qubits) in the quantum computer

• The The function function expexp is the traditional exponential function, but some is the traditional exponential function, but some care must be taken here because the care must be taken here because the argument is a matrixargument is a matrix. .

• The evolution matrix is the The evolution matrix is the programprogram for the quantum for the quantum computer. Applying this program to the input state produces the computer. Applying this program to the input state produces the output state ,which gives us a solution to the problem.output state ,which gives us a solution to the problem.

0ˆ0ˆexp0

tUdHiTt

t

t

0 !exp

n

n

n

xx

tU

We discussed this power series

already

Formula from last

slide

The Hamiltonian Matrix in Schroedinger EquationThe Hamiltonian Matrix in Schroedinger Equation

• The Hamiltonian is a The Hamiltonian is a matrixmatrix that tells us that tells us how the quantum how the quantum computer reacts computer reacts to the application of signals. to the application of signals.

• In other words, it describes how the qubits behave under the In other words, it describes how the qubits behave under the influence of a influence of a machine languagemachine language consisting of varying consisting of varying some some controllable parameterscontrollable parameters (like electric or magnetic fields). (like electric or magnetic fields).

• Usually, the form of the Usually, the form of the matrix Hmatrix H needs to be either needs to be either derived derived by a physicistby a physicist or or obtained via direct measurementobtained via direct measurement of the of the properties of the computer.properties of the computer.

0ˆ0ˆexp0

tUdHiTt

t

The Evolution Matrix in the Schrodinger EquationThe Evolution Matrix in the Schrodinger Equation

• While the Hamiltonian describes how the quantum computer While the Hamiltonian describes how the quantum computer responds to the machine language, responds to the machine language, the the evolution matrixevolution matrix describes describes the effect that this has on the state of the quantum computer.the effect that this has on the state of the quantum computer.

• While While knowing the Hamiltonian allows us to calculate the knowing the Hamiltonian allows us to calculate the evolution matrix in a pretty straightforward wayevolution matrix in a pretty straightforward way, , the reverse is the reverse is not true.not true.

• If we know the If we know the programprogram, , by which is meant the evolution by which is meant the evolution matrixmatrix, it is not an easy problem to determine the machine , it is not an easy problem to determine the machine language sequence that produces that program. language sequence that produces that program.

• This is the quantum computer science version of the This is the quantum computer science version of the compilercompiler problemproblem. Or “. Or “logic synthesislogic synthesis” problem.” problem.

0ˆ0ˆexp0

tUdHiTt

t

Evolution matrixEvolution matrixHamiltonian matrixHamiltonian matrix

Postulate 3:Postulate 3: Quantum Quantum

MeasurementMeasurement

Computational Basis – a reminderComputational Basis – a reminder

We recalculate to a new basis

Observe that it is not required to be orthonormal,

just linearly independent

1/2

Example of measurement Example of measurement in different bases in different bases

The second with probability zero

• You can check from definition that inner product You can check from definition that inner product of |0> and |1> is zero.of |0> and |1> is zero.

• Similarly the inner product of vectors from the Similarly the inner product of vectors from the second basis is zero.second basis is zero.

• But we can take vectors like |0> and 1/But we can take vectors like |0> and 1/2(|0>-|2(|0>-|1>) as a basis also, although measurement will 1>) as a basis also, although measurement will perhaps suffer.perhaps suffer.

Good base Not a base

A simplified Bloch Sphere to illustrate A simplified Bloch Sphere to illustrate the bases and measurementsthe bases and measurements

You cannotYou cannot add more vectors that would be orthogonal together with blue or red vectors

Probability and MeasurementProbability and Measurement• A A yes/no measurementyes/no measurement is an is an interactioninteraction designed to designed to

determine whether a given system is in a certain state determine whether a given system is in a certain state ss..

• The amplitude of state The amplitude of state ss, given the actual state , given the actual state tt of the of the system determines the system determines the probabilityprobability of getting a “yes” of getting a “yes” from the measurement.from the measurement.

• Important:Important: For a system prepared in state For a system prepared in state tt, , anyany measurement that asks “is it in state measurement that asks “is it in state ss?” will return “yes” ?” will return “yes” with probability Pr[with probability Pr[ss||tt] = |] = |ss††tt||22

– After the measurement, After the measurement, the state is changedthe state is changed, in a way we will , in a way we will define later.define later.

Inner product between states

A Simple Example of distinguishable, non-A Simple Example of distinguishable, non-distinguishable states and measurementsdistinguishable states and measurements

• Suppose abstract system Suppose abstract system SS has a set of only 4 has a set of only 4 distinguishable possible statesdistinguishable possible states, which we’ll call , which we’ll call ss00, , ss11, , ss22, and , and ss33, with , with corresponding ket vectorscorresponding ket vectors | |

ss00, |, |ss11, |, |ss22, and |, and |ss33..• Another possible stateAnother possible state is then the vector is then the vector

• Which is equal to the Which is equal to the column matrixcolumn matrix::• If measuredIf measured to see if it is in state to see if it is in state ss00,,

we have a 50% chance of getting a “yes”. we have a 50% chance of getting a “yes”.

3022

1ss

i

2

0

0

21

i

ObservablesObservables

ObservablesObservables• Hermitian operator Hermitian operator AA on on VV is called an is called an

observableobservable if there is an if there is an orthonormalorthonormal (all unit- (all unit-length, and mutually orthogonal) subset of its length, and mutually orthogonal) subset of its eigenvectors that forms a eigenvectors that forms a basisbasis of of VV..

Observe that the

eigenvectors must be

orthonormal

There can be measurements that are not observables

ObservablesObservables• Postulate 3:Postulate 3:

– Every measurable physical property of a system is Every measurable physical property of a system is described by a corresponding described by a corresponding operator operator A.A.

– Measurement outcomes correspond to Measurement outcomes correspond to eigenvalues.eigenvalues.

• Postulate 3a:Postulate 3a: – The probability of an outcome is given by the The probability of an outcome is given by the

squared absolute amplitudesquared absolute amplitude of the corresponding of the corresponding eigenvector(s), given the state.eigenvector(s), given the state.

Density Density OperatorsOperators

Density OperatorsDensity Operators• For a given state |For a given state |, the , the probabilities of all the probabilities of all the

basis states basis states ssii are determined by an are determined by an Hermitian Hermitian

operatoroperator or or matrix matrix (the (the densitydensity matrix): matrix):

• The The diagonal elementsdiagonal elements ii,,ii are the are the probabilities of probabilities of

the basis states.the basis states.– The off-diagonal elements are The off-diagonal elements are “coherences”.“coherences”.

• The density matrix describes the state exactly.The density matrix describes the state exactly.

nnn

n

ijji

cccc

cccc

cc**

1

1*

1*1

*, ][][

M =M =

Hermitian Hermitian

observableobservable

Towards QM Postulate 3 on Towards QM Postulate 3 on measurement and general formulasmeasurement and general formulas

A A measurementmeasurement is described by an is described by an Hermitian Hermitian operator (operator (observableobservable))

MM = = mm PPmm

– PPmm is the is the projectorprojector onto the eigenspace of onto the eigenspace of MM with with

eigenvalue eigenvalue mm– After the measurement the state will be After the measurement the state will be with with

probability p(probability p(mm) = ) = ||PPmm||..– e.g. measurement of a qubit in the computational basise.g. measurement of a qubit in the computational basis

• measuring |measuring | = = |0|0 + + |1|1 gives: gives:• |0|0 with probability with probability |0|00|0| = | = |0|0|||22 = | = |||22

• |1|1 with probability with probability |1|11|1| = | = |1|1|||22 = | = |||22

m

Pm|

p(m)

eigenvalue

Derived in next slide

Hermitian Hermitian observableobservable

How to survive reading papersHow to survive reading papers

1.1. In all these calculations we have a “ket part” and a “bra In all these calculations we have a “ket part” and a “bra part”part”

2.2. Remember how to create Bra part from Ket Part.Remember how to create Bra part from Ket Part.3.3. Remember about dual vectorsRemember about dual vectors4.4. Remember about inner products, how to calculate themRemember about inner products, how to calculate them5.5. Remember that there are many rules to operate on these Remember that there are many rules to operate on these

vectors so you do not have to do all calculations on vectors so you do not have to do all calculations on Heisenberg matrices.Heisenberg matrices.

6.6. If you are confused with notations, always convert to If you are confused with notations, always convert to Heisenberg matrices.Heisenberg matrices.

An example how An example how Measurement OperatorsMeasurement Operators act act on the state space of a quantum systemon the state space of a quantum system

0

1

1

2

1

2

100opH

000 M111 M

2

1

1

1

2

1

00

0111

2

100

2

1

1

1

2

1

10

0011

2

111

Measurement operators act on the state space of a quantum systemInitial state:

Operate on the state space with an operator that preservers unitary evolution:

Define a collection of measurement operators for our state space:

Act on the state space of our system with measurement operators:

Hadamard

Half probability of measuring 0

Half probability of measuring 1

But now it is a part of formal calculus

= p= p00

= = |0|0++|1|1

Qubit exampleQubit example: calculate the : calculate the density density matrixmatrix

Suppose 0 with probability 1. 1 1 0

Then 0 0 1 0 .0 0 0

0 0 0Then 1 1 0 1 .

1 0 1

Suppose 1 with probability 1.

0 1Suppose with probability 1.

2

i

1 10 1 0 1 1 1Then 1 .

12 22 2

ii ii

i i

Conjugate and change kets to bras

where is th density matre .ixj j jj

p Density matrix

Density matrix is a generalization of state

Measurement of a state vector using Measurement of a state vector using projective measurementprojective measurement

0

1

1

2

1

2

100opH

01

1

2

1

10

0111

2

1

10

01

11

1

2

1

01

1011

2

1

01

10

01

1

2

1

0

011

2

1

0

0

i

i

i

i

Operate on the state space with an operator that preservers unitary evolution:

Define observables:

Act on the state space of our system with observables (The average value of measurement outcome after lots of

measurements):

10

01z

01

10x

0

0

i

iy

This type of measurement represents the limit as the number of measurements goes to infinity

Here 3 may be enough, in general you need four

Vector is on axis X

Bloch SphereBloch Sphere

Very General FormulaVery General Formula• probability p(probability p(mm) = ) = ||PPmm||..

Any type of measurement

operator

““insert Operator insert Operator between Braket” rulebetween Braket” rule

PPmm|| = state (column vector) = state (column vector)

||PPmm|| = number = probability = number = probability

This is inner product not tensor product!

More examples and generalizations: Duals and More examples and generalizations: Duals and Inner Products are used in measurementsInner Products are used in measurements

<<||

( )

We prove from general properties of operators

Remember this is a number

To do bra from ket you need transpose and conjugate to make a row vector of conjugates.

Do this always if you are in trouble to understand some formula involving operators, density matrices, bras and kets.

Review:Review: Duals as Row Vectors Duals as Row Vectors

General General MeasurementMeasurement

To prove it it is sufficient to substitute the old base and calculate, as shown

Review:Review:

Illustration of the formalisms used. You Illustration of the formalisms used. You can calculate measurements from therecan calculate measurements from there

12

sin02

cos ie

0

1

cossin

sincos)(

i

iX

i

i

e

eZ

0

0

cossin

sincos)(Y

10

01z

01

10x

0

0

i

iy

10

**

**

t

t

e

e

State Vector

Density State

This is calculate as in 2 slides earlier

Postulate 3, rough formPostulate 3, rough form

Ensamble Quantum Ensamble Quantum Computers and Computers and

Mixed StatesMixed StatesParticle == qubit

Molecule == set of qubits = single

quantum computer

Very many identical

Molecules == EnsambleEnsamble quantum

computer

Mixed StatesMixed States• Suppose one only knows of a system that it is in one of a Suppose one only knows of a system that it is in one of a statistical statistical

ensembleensemble of state vectors of state vectors vvii (“pure” states), each with (“pure” states), each with density density

matrix matrix ii and and probability Pprobability Pii..

• This is called a This is called a mixed statemixed state..

• This ensembleThis ensemble is is completely described, for all physical purposescompletely described, for all physical purposes,, by the by the expectation valueexpectation value ( (weighted averageweighted average) of density matrices:) of density matrices:

– Note: even if there were Note: even if there were uncountablyuncountably many many state vectors state vectors vvii, the state remains fully , the state remains fully

described by <described by <nn22 complex numbers, where complex numbers, where nn is the number of basis states! is the number of basis states!

iiP

Ensemble Ensemble of quantum statesof quantum states• Quantum states can be expressed as a Quantum states can be expressed as a density density

matrix:matrix:

• A system with A system with nn quantum states has quantum states has nn entries across entries across the diagonal of the density matrix. The the diagonal of the density matrix. The nnthth entry of entry of the diagonal corresponds to the the diagonal corresponds to the probability of the probability of the systemsystem being measured in the being measured in the nnthth quantum state. quantum state.

• The The off diagonal correlationsoff diagonal correlations are zeroed out by are zeroed out by decoherence.decoherence. TUU *

i

iiip For the ensemble

Sum of these probabilities is oneSum of these probabilities is one

Unitary operations on a density matrixUnitary operations on a density matrix

• Unitary operations on a Unitary operations on a density matrixdensity matrix are are expressed as:expressed as:

• In other words the diagonal of density matrix is left In other words the diagonal of density matrix is left as as weightsweights corresponding to the corresponding to the current states current states projectionprojection onto the onto the computational basiscomputational basis after acted on after acted on by the unitary operator by the unitary operator UU– Much like an inner product. like an inner product.

TUU *

UUUUp i

i

ii

Old density matrix

New density matrix

adjoint

i

iiip

Trace of Trace of MatrixMatrix

Trace of MatrixTrace of Matrix• Trace of a matrix (sum of the diagonal elements):Trace of a matrix (sum of the diagonal elements):

• Unitary operators are Unitary operators are trace preservingtrace preserving. .

• The trace of a The trace of a pure statepure state is 1, all information about the system is is 1, all information about the system is known.known.

• Operators Commute under the action of the trace: Operators Commute under the action of the trace:

i

iiAAtr )(

)()( YXtrXYtr

TUU *

The trace operationThe trace operation tr j jjA A

0 1 1 0

tr 0; trExamples 21 0

: .0 1

X X I I

trj jj

AB AB

Cyclicity proper try: r .t =tAB BA

jk kjjkA B kj jkjk

B A kkkBA tr BA

Partial Trace of a MatrixPartial Trace of a Matrix• Partial Trace (defined by linearity)Partial Trace (defined by linearity)• If you want to know about the If you want to know about the nnthth state in a system, you state in a system, you

can can trace over the other statestrace over the other states..

i

iiAAtr )(

)( )( 21212121 bbtraabbaatrB

)( ABBtr

TUU *

Partial trace

Density matrix trace

Measurement of a Measurement of a density state for density state for

circuit with circuit with entanglemententanglement

Measurement of a density state for Measurement of a density state for circuit with entanglementcircuit with entanglement

Initial state:

0000

0000

0000

0001

0000

Operate on the state space with an operator that preservers unitary evolution (H gate first bit):

1111 0000' UUIHIH

Now act on system with CNOT gate:

1001

0000

0000

1001

2

1' 1,21,2121112 UUCNOTUUCNOT

00000 M 01011 M 10102 M 11113 M

We still define collections of measurement operators to act on the state space of our system:

H

Measurement of a density stateMeasurement of a density state

)( )( 21212121 bbtraabbaatrB tr kP

Probability of outcome k

The probability that a result m occurs is given by the equation:

2

1

1001

0000

0000

1001

2

1

1000

0000

0000

0000

)(

trMtrMMtrmp mmm

For most of our purposes we can just use state vectors.

M3

tr kPrecall

0000

0000

0000

1001

½ tr =1/2

REMINDER: Ensemble point of viewREMINDER: Ensemble point of view

I magine that a quantum system is in the state withprobability .

j

jp

We do a measurement described by projectors .kP

Probability of outcome Pr | state j jk

k k p j j jk

k

P p trj j j k

k

p P

Probability of being in state j

Probability of outcome k being in state j

Formula linking

trace and density matrix

REMINDER: Ensemble point of viewREMINDER: Ensemble point of view

Probability of outcome Pr | state j jk

k k p j j jk

k

P p

trj j j kk

p P

where is th density matre .ixj j jj

p tr kP

all mecomple asuremtely determi ent statistnes ics.

Probability of outcome k

Probability of being in state j

Probability of outcome k being in state j

Postulate 3:Postulate 3: Quantum Quantum

MeasurementMeasurement

Postulate 3:Postulate 3: Quantum MeasurementQuantum Measurement

Now we can formulate precisely the Postulate 3

Now we use this notation for Now we use this notation for an Example of Qubit an Example of Qubit

MeasurementMeasurement

1 0 0 0

1 0 0 0

[0* 1

*]0

1

1 0 0 0

= [0* 1

*]

0

1

We show here two methods to derive

it. Check for homeworks and

exam

How the state vector changes in How the state vector changes in measurement?measurement?

M0 |> =

= |0> (<0| 0 |0> + <0| 1 |1> = |0> 0

What happens to a system after a What happens to a system after a Measurement?Measurement?

• After a system or subsystem is measured from outside, its state After a system or subsystem is measured from outside, its state appears to appears to collapsecollapse to exactly match the measured outcometo exactly match the measured outcome

– the amplitudes of all states perfectly distinguishable from states consistent with the amplitudes of all states perfectly distinguishable from states consistent with that outcome that outcome drop to zerodrop to zero

– states consistent with measured outcome can be considered states consistent with measured outcome can be considered “renormalized”“renormalized” so so their probabilities their probabilities sum to 1sum to 1

Only orthogonal state can be distinguished in a measurement

Projective Measurements: Projective Measurements: Average Values and Standard Average Values and Standard

DeviationsDeviationsObservable:

Can write:

Average value of a measurement:

Standard deviation of a measurement:

scalar

scalar

scalar

PhasePhaseGlobal phaseGlobal phase

Relative phase can be physically distinquished

Relative phase depends on the basis

Postulate 4:Postulate 4: Composite Composite

SystemsSystems

Compound SystemsCompound Systems• Let Let C=ABC=AB be a system composed of two be a system composed of two

separate subsystems separate subsystems A, BA, B each with vector each with vector spaces spaces AA, , BB with bases | with bases |aaii, |, |bbjj..

• The state space of C is a vector space The state space of C is a vector space CC==AABB given by the given by the tensor producttensor product of spaces of spaces AA and and B,B, with basis states with basis states labeled as |labeled as |aaiibbjj..

Composition exampleComposition example

The state space of a composite physical system is The state space of a composite physical system is the the tensor product of the state spaces of the tensor product of the state spaces of the componentscomponents

– nn qubits represented by a 2 qubits represented by a 2nn-dimensional Hilbert space-dimensional Hilbert space– composite state is |composite state is | = | = |11 | |22 . . .. . . | |nn– e.g. 2 qubits:e.g. 2 qubits:

||11 = = 11|0|0 + + 11|1|1||22 = = 22|0|0 + + 22|1|1|| = | = |11 | |22 = = 1122|00|00 + + 1122|01|01 + + 1122|10|10 + + 1122|11|11

– entanglemententanglement2 qubits are2 qubits are entangledentangled if | if | | |11 | |22 for any | for any |11, |, |22e.g. |e.g. | = = |00|00 + + |11|11

EntanglementEntanglement• If the state of compound system C If the state of compound system C cancan be expressed be expressed as a as a

tensor product of states of two independent subsystems A tensor product of states of two independent subsystems A and B,and B,

cc = = aabb,,

• then, we say that A and B are then, we say that A and B are not entanglednot entangled, and they , and they have have individual statesindividual states..– E.g. |00E.g. |00+|01+|01+|10+|10+|11+|11=(|0=(|0+|1+|1))(|0(|0+|1+|1))

• Otherwise, A and B are Otherwise, A and B are entangledentangled (basically correlated); (basically correlated); their states are not independent.their states are not independent.– E.g. |00E.g. |00+|11+|11

Entanglement results from axioms/postulates. It exists only

for quantum states

Size of Compound State SpacesSize of Compound State Spaces• Note that a system composed of many separate Note that a system composed of many separate

subsystems has a subsystems has a very largevery large state space state space..

• Say it is composed of Say it is composed of NN subsystems, each with subsystems, each with kk basis basis states:states:

– The compound system The compound system has has kkNN basis basis states! states!– There are states of the compound system having There are states of the compound system having nonzero nonzero

amplitude in amplitude in allall these these kkNN basis states basis states!!

– In such states, all the distinguishable basis states are In such states, all the distinguishable basis states are (simultaneously) possible outcomes (each with some (simultaneously) possible outcomes (each with some corresponding probability)corresponding probability)

– Illustrates the Illustrates the “many worlds”“many worlds” nature of quantum mechanics. nature of quantum mechanics.

Postulate 4:Postulate 4: Composite SystemsComposite Systems

Summary on Postulates Summary on Postulates

Hilbert Space

Evolution

Measurement

Tensor Product

Key Points to Remember:Key Points to Remember:• An abstractly-specified system may have many possible An abstractly-specified system may have many possible

states; only some are states; only some are distinguishabledistinguishable..• A A quantum state/vector/wavefunctionquantum state/vector/wavefunction assigns a complex- assigns a complex-

valued valued amplitudeamplitude ((ssii) to each distinguishable state ) to each distinguishable state ssii (out (out

of some basis set)of some basis set)• The The probabilityprobability of state of state ssii is | is |((ssii)|)|22, the square of , the square of ((ssii)’s )’s

length in the complex plane.length in the complex plane.• States evolveStates evolve over time via unitary (invertible, length- over time via unitary (invertible, length-

preserving) transformations.preserving) transformations.• Statistical mixtures of statesStatistical mixtures of states are represented by are represented by weighted weighted

sumssums of density matricesof density matrices =|=||.|.

Key points to Key points to remember remember

• The Schrödinger EquationThe Schrödinger Equation• The HamiltonianThe Hamiltonian• The Evolution MatrixThe Evolution Matrix• How complicated is a single Quantum Bit?How complicated is a single Quantum Bit?• Measurement Measurement • Measurement operators Measurement operators • Measurement of a state vector using projective Measurement of a state vector using projective

measurementmeasurement• Density Matrix and the TraceDensity Matrix and the Trace• Ensembles of quantum states, basic definitions and Ensembles of quantum states, basic definitions and

importanceimportance• Measurement of a density stateMeasurement of a density state

Bibliography & acknowledgementsBibliography & acknowledgements• Michael A. Nielsen and Isaac L. Chuang, Quantum

Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2002

• V. Bulitko, On quantum Computing and AI, Notes for a graduate class, University of Alberta, 2002

• R. Mann,M.Mosca, Introduction to Quantum Computation, Lecture series, Univ. Waterloo, 2000 http://cacr.math.uwaterloo.ca/~mmosca/quantumcoursef00.htm D. Fotin, Introduction to “Quantum Computing Summer School”, University of Alberta, 2002.

Additional Additional SlidesSlides

DistinguishabilityDistinguishability

On the other handThus we have contradiction, states can be distinguished unless they are orthogonal

Recall that M is

measurement operator

We represent |2 in new base

General Measurements General Measurements in compound spacesin compound spaces

Uncertainty Uncertainty PrinciplePrinciple

Positive Operator-Valued Positive Operator-Valued Measurements (POVM)Measurements (POVM)

• This lecture was taught in 2005, 2007, 2011This lecture was taught in 2005, 2007, 2011• There is more about trace and partial trace in There is more about trace and partial trace in

future lectures and examples.future lectures and examples.