Quantum InformationStephen M. Barnett

University of Strathclyde

steve@phys.strath.ac.uk

The Wolfson Foundation

1. Probability and Information

2. Elements of Quantum Theory

3. Quantum Cryptography

4. Generalized Measurements

5. Entanglement

6. Quantum Information Processing

7. Quantum Computation

8. Quantum Information Theory

4.1 Ideal von Neumann measurements4.2 Non-ideal measurements4.3 Probability operator measures4.4 Optimised measurements4.5 Operations

Measurement model

What are the probabilities for the measurement outcomes?How does the measurement change the quantum state?

‘Black box’

4.1 Ideal von Neumann measurements

Red

von Neumann measurements

Mathematical Foundations of Quantum Mechanics

(i) An observable A is represented by an Hermitian operator:

(ii) A measurement of A will give one of its eigenvalues as a result. The probabilities are

or

(iii) Immediately following the measurement, the system is left in the associated eigenstate:

nnnA ˆ

nnnnP 2

)(

ˆTrˆ)( nnnnnP

nn ˆ

More generally for observables with degenerate spectra:

(i’) Let be the projector onto eigenstates with eigenvalue :

(ii’) The probability that the measurement will give the result is

(iii’) The state following the measurement is

nP n

n

nnn PPA ˆˆˆ

ˆˆTr)( nn PP

)ˆˆ(Tr

ˆˆˆˆ

n

nn

PPP

Detector

a b

DetectdetectNo

Detect

ˆ-Iˆ

ˆ

PP

dxxxPb

a

Properties of projectors

I. They are Hemitian Observable

II. They are positive Probabilities

III. They are complete Probabilities

IV. They are orthonormal ??

nn PP ˆˆ †

0nP

n

nP Iˆ

iijji PPP ˆˆˆ

4.2 Non-ideal measurements

Real measurements are ‘noisy’ and this leads to errors

‘Black box’

‘Black box’

0

1

pP 1)0(

pP )1(

pP )0(

pP 1)1(

For a more general state:

0ˆ01ˆ1)1()1(

1ˆ10ˆ0)1()0(

ppP

ppP

We can write these in the form

)ˆˆ(Tr)1()ˆˆ(Tr)0(

1

0

PP

0011)1(ˆ

1100)1(ˆ

1

0

pp

pp

where we have introduced the probability operators

These probability operators are

I. Hermitian

II. positive

III. complete

But

IV. they are not orthonormal

1,0†

1,0 ˆˆ

001)1(ˆ

010)1(ˆ22

1

22

0

pp

pp

I1100ˆˆ 10

0I)1(ˆˆ 10 pp

We seem to need a generalised description of measurements

4.3 Probability operator measures

Our generalised formula for measurement probabilities is

The set probability operators describing a measurement is called a probability operator measure (POM) or a positiveoperator-valued measure (POVM).

The probability operators can be defined by the properties that they satisfy:

ˆˆTr)( iiP

Properties of probability operators

I. They are Hermitian Observable

II. They are positive Probabilities

III. They are complete Probabilities

IV. Orthonormal ??

nn ˆˆ †

0ˆn

n

n I

iijji ˆˆˆ

Generalised measurements as comparisons

S A

S A

S + A

Prepare an ancillary system ina known state:

Perform a selected unitary transformation to couple the systemand ancilla:

Perform a von Neumann measurementon both the system and ancilla:

AS

AU S ˆ

iiS Ai

The probability for outcome i is

The probability operators act only on the system state-space.

AUiiUA

AUiiUA

ˆˆ

ˆˆ

†

††

0ˆˆ2 AUi Si

A,Si

ii I

POM rules: I. Hermiticity:

II. Positivity:

III. Completeness follows from:

SS

SS

AUiiUA

iUAAUiiP

ˆˆ

ˆˆ)(†

†

i

i

Generalised measurements as comparisons

We can rewrite the detection probability as

is a projector onto correlated (entangled) states of the system and ancilla. The generalised measurement is a von Neumann measurement in which the system and ancilla are compared.

APAiP SiS ˆ)(

UiiUPiˆˆˆ †

APA iiˆˆ

0ˆˆˆˆ APAAPA mnmn

Simultaneous measurement of position and momentum

The simultaneous perfect measurement of x and p would violatecomplementarity.

x

p Position measurement gives nomomentum information and depends on the position probabilitydistribution.

Simultaneous measurement of position and momentum

The simultaneous perfect measurement of x and p would violatecomplementarity.

x

p

Momentum measurement gives noposition information and depends on the momentum probabilitydistribution.

Simultaneous measurement of position and momentum

The simultaneous perfect measurement of x and p would violatecomplementarity.

x

p Joint position and measurement gives partial information on both the position and the momentum.

Position-momentum minimumuncertainty state.

POM description of joint measurements

Probability density:

),(ˆˆ),( mmmm pxTrpx

Minimum uncertainty states:

I,,21

4)(exp2, 2

24/12

mmmmmm

mmmm

pxpxdpdx

xxpixxdxpx

This leads us to the POM elements:

mmmmmm pxpxpx ,,21),(ˆ

The associated position probability distribution is

2

22

22

2

2

4)(Var&

)(Var

2)(expˆ)(

pp

xx

xxxxdxx

m

m

mm

Increased uncertainty is the price we pay for measuring x and p.

The communications problem

‘Alice’ prepares a quantum system in one of a set of N possiblesignal states and sends it to ‘Bob’

Bob is more interested in

ijijP ˆˆTr)|(

)ˆˆ(Tr

ˆˆTr)|(

j

iij pjiP

Preparationdevice

i selected.

prob. ip

i Measurement

result j

Measurementdevice

In general, signal states will be non-orthogonal. No measurement can distinguish perfectly between such states.

Were it possible then there would exist a POM with

Completeness, positivity and

What is the best we can do? Depends on what we mean by ‘best’.

121212

222111

ˆ0ˆ

ˆ1ˆ

1ˆ 111

0ˆandpositiveˆˆˆ 1111 AAA

0ˆˆ 22122

221212 A

Minimum-error discrimination

We can associate each measurement operator with a signalstate . This leads to an error probability

Any POM that satisfies the conditions

will minimise the probability of error.

ii

jjj

N

je TrpP ˆˆ1

1

jpp

kjpp

jj

N

kkkk

kkkjjj

0ˆˆˆ

,0ˆ)ˆˆ(ˆ

1

For just two states, we require a von Neumann measurement withprojectors onto the eigenstates of with positive (1) and negative (2) eigenvalues:

Consider for example the two pure qubit-states

2211 ˆˆ pp

221121min ˆˆTr1 ppPe

1sin0cos

1sin0cos

2

1

)2cos(21

1

1

2

2

The minimum error is achieved by measuring in the orthonormal basis spanned by the states and .

We associate with and with :

The minimum error is the Helstrom bound

12

1 1 2 2

2

122

2

211 ppPe

2/1221212

1min 411 ppPe

+

P = ||2

P = ||2

A single photon only gives one “click”

But this is all we need to discriminate between our two states with minimum error.

A more challenging example is the ‘trine ensemble’ of threeequiprobable states:

It is straightforward to confirm that the minimum-error conditionsare satisfied by the three probability operators

31

33

31

221

2

31

121

1

0

130

130

p

p

p

iii 32ˆ

Simple example - the trine states

Three symmetric states of photon polarisation

3

23

2

23

1

Minimum error probabilityis 1/3.

This corresponds to a POMwith elements

ˆ j 23 j j

How can we do a polarisationmeasurement with these threepossible results?

Polarisation interferometer - Sasaki et al, Clarke et al.

PBS

PBS

PBS

2

2

2/32/1

2/32/1

01

2/3

02/3

000

10

1/ 20

1/ 20

6/112/1

6/112/1

3/23/1

06/1

03/2

06/1

3/2

06/1

06/1

0

6/1

06/1

03/2

0

Unambiguous discrimination

The existence of a minimum error does not mean that error-freeor unambiguous state discrimination is impossible. A von Neumannmeasurement with

will give unambiguous identification of :

result error-free

result inconclusive

2

111111ˆˆ PP

21

?1

There is a more symmetrical approach with

21?

1121

2

2221

1

ˆˆIˆ

11ˆ

11ˆ

Result 1 Result 2 Result ?

State 0

State 0

211

2

1

211

21

21

How can we understand the IDP measurement?

Consider an extension into a 3D state-space

a

ba

b

Unambiguous state discrimination - Huttner et al, Clarke et al.

a

b

?a

b

A similar device allows minimum error discriminationfor the trine states.

Measurement model

What are the probabilities for the measurement outcomes?How does the measurement change the quantum state?

‘Black box’

Red

How does the state of the system change after a measurement is performed?

Problems with von Neumann’s description:

1) Most measurements are more destructive than von Neumann’s ideal.

2) How should we describe the state of the system after a generalised measurement?

4.5 Operations

Physical and mathematical constraints

What is the most general way in which we can change a density operator?

Quantum theory is linear so …

or more generally

BUT this must also be a density operator.

BA ˆˆˆˆ

iii

BA ˆˆˆˆ

Properties of density operators

I. They are Hermitian

II. They are positive

III. They have unit trace

ˆˆ †

0ˆ

1ˆTr

The first of these tells us that and this ensures that the second is satisfied.

The final condition tells us that

†ˆˆii AB

Iˆˆ † iii

AA

The operator is positive and this leads us to associate

(Knowing does not give us .)If the measurement result is i then the density operator changesas

This replaces the von Neumann transformation

ii AA ˆˆ †

iii AA ˆˆˆ †

)ˆˆ(Tr

ˆˆˆ

)ˆˆˆ(Tr

ˆˆˆˆ

†

†

†

i

ii

ii

ii AAAAAA

)ˆˆ(Tr

ˆˆˆˆ

n

nn

PPP

iAi

If the measurement result is not known then the transformed density operator is the probability-weighted sum

which has the form required by linearity.

We refer to the operators and as Krauss operatorsor an an ‘effect’.

††

†† ˆˆˆ

)ˆˆˆ(Tr

ˆˆˆ)ˆˆˆ(Trˆ ii

iii

ii

iii AA

AAAAAA

iA †ˆiA

Repeated measurements

Suppose we perform a first measurement with results i and effectoperators and then a second with outcomes j and effects .

The probability that the second result is j given that the first was i is

iA jB

)ˆˆˆˆˆ(Tr)()|(),(

)ˆˆˆ(Tr

)ˆˆˆˆˆ(Tr)|(

††

†

††

ijji

ii

iijj

ABBAiPijPjiP

AA

AABBijP

Hence the combined probability operator for the two measurementsis

If the results i and j are recorded then

If they are not known then

ijjiij ABBA ˆˆˆˆˆ ††

),(

ˆˆˆˆˆˆ

††

jiPBAAB jiij

††

,

ˆˆˆˆˆˆ jiijji

BAAB

Unitary and non-unitary evolution

The effects formalism is not restricted to describing measurements,e.g. Schroedinger evolution

which has a single Krauss operator .

We can also use it to describe dissipative dynamics:

/ˆexp)0(ˆ/ˆexp)(ˆ)0(ˆ tHitHit

/ˆexpˆ tHiA

e

g

Spontaneous decay rate 2

We can write the evolved density operator in terms of two effects:

Measurement interpretation?

Has the atom decayed?

egetA

ggeeetA

AtAAtAt

tY

tN

YYNN

2

††

1)(ˆ

)(ˆ

ˆ)0(ˆ)(ˆˆ)0(ˆ)(ˆ)(ˆ

eeeAA

ggeeeAAt

YYY

tNNN

)1(ˆˆˆ

ˆˆˆ2†

2†

e

g

e

g

No detection

122

)(ˆ

gg

tee

ggt

ge

teg

tee

ggge

egee ee

ee

e

gDetection

gg

Optimal operations: an example

What processes are allowed? Those that can be described by effects.

State separation

Suppose that we have a system known to have been preparedin one of two non-orthgonal states and . Our taskis to separate the states, i.e. to transform them into and with

so that the states are more orthogonal and hence more distinguishable. This process cannot be guaranteed but can succeed with some probability . How large can this be?

1

1

2

2

1122

SP

We introduce an effect associated with successful state separation

We can bound the success probability by considering the action of on a superposition of the states and noting that theresult must be an allowed state:

221 ||ˆ SS PA

1SA

22

112

1

1||

SP

There is a natural interpretation of this in terms of unambiguousstate discrimination:

Because this is the maximum allowed, state separation cannot better it so

)2(1)2(1)2(1?

)2(1Conc 11 PP

22

11

1Conc

2Conc

1

1

S

S

P

PPP

ssAs

A

A

clone

AAA

s

No cloning theorem - Wootters & Zurek, Dieks

Can we copy an unknown state ?Suppose it is possible:

But the superposition principle then gives:

Exact cloning? - Duan & Guo, Chefles

Ab

Aa bb

aa

?

Pclone

1 Pclone

Error-free discrimination probability

For the cloned states

baP 1IDP

ba

bbaaP

1

1clonedIDP

Cloning cannot increase the discrimination probability

baP

PPP

11

clone

clonedIDPcloneIDP

separation bound

Can we clone exactly a quantum system known to be in the state |a or |b?

Summary

• The projective von Neumann measurements do not provide the most general description of a measurement.

• The use of probability operators (POMs) allows us to seek optimal measurements for any given detection problem.

• The language of operations and effects allows us to describe, in great generality, the post-measurement state.

Naimark’s theorem

All POMs correspond to measurements. Consider a POM for aqubit with N probability operators:

POM conditions:

Can we treat this as a von Neumann measurement in a enlarged state-space?

10,ˆ 10 jjjjjj

N

jjj

N

jjj

N

jj

N

jj

11

*0

10

*1

1

2

11

2

0

0

1

Our task is to represent the vectors as the components in the qubit space of a set of orthonormal states in an enlarged space.

Enlarged space spanned by N orthornormal states .

The extra states can be other states of the system or by introducing an ancilla:

An alternative orthonormal basis is

jj

j

jUjN

jjii

ˆ1

*

mn

N

jjnjmnm

1

*

22100 0,,0,0,1,0 NAAAAA

Because is unitary, we can also form the orthonormal basis:

These are the required orthonormal states for our von Neumannmeasurement:

All measurements can be described by a POM and all POMs describe possible measurements.

U

ijU ji

N

ij

1

0

†ˆ

ˆˆTr

ˆ

ˆ)(

j

jj

jjjP

Summary

The probabilities for the possible outcomes of any given measurement can be written in the form:

iTriP ˆˆ)(

The probability operators satisfy the three conditions

Iˆ,0ˆ,ˆˆ † ii

iii

All measurements can be described in this way and all sets of operators with these properties represent possible measurements.

Poincaré Sphere

Optical polarization

Bloch Sphere

Electron spin

Spin and polarisation QubitsPoincaré and Bloch Spheres

Two state quantum system

States of photon polarisation

Horizontal

Vertical

Diagonal up

Diagonal down

Left circular

Right circular

0

1

102

1

102

1

102

1 i

102

1 i

Example from quantum optics:How can we best discriminate (without error) between the coherent states and ?

Coherent states:

Interfere like classical amplitudes.

Symmetric beamsplitter

r t

r t

nn

n

n !0

Unambiguous discrimination between coherent states - Hutner et al

i

0or2)(2 2/12/1 iii

2/12/1 2or0)11(2

This interference experiment puts all the light into a single output.Inconclusive results occur when no photons are detected. This happens with probability

)||2exp(02 222/1?P