Photons and Quantum InformationStephen M. Barnett

1. A bit about photons

2. Optical polarisation

3. Generalised measurements

4. State discriminationMinimum errorUnambiguousMaximum confidence

1. A bit about photons

Photoelectric effect - Einstein 1905

WheV

BUT …

Modern interpretation: resonance with the atomic transition frequency.

We can describe the phenomenon quantitatively by a model in which the matter is described quantum mechanically but the light is described classically.

Photons?

Single-photon (?) interference - G. I. Taylor 1909

Light source Smoked glass screens Screen with two slits

Photographic plate

Longest exposure - three months

“According to Sir J. J. Thompson, this sets a limit on the size of the indivisible units.”

Single photons (?) Hanbury-Brown and Twiss

1

2 1)0(

)2()1(

)2,1(

2

2)2(

2

I

Ig

ITPIRP

IRTTIRIP

Blackbody light g(2)(0) = 2

Laser light g(2)(0) = 1

Single photon g(2)(0) = 0 !!!

violation of Cauchy-Swartz inequality

Single photon source - Aspect 1986

Detection of the first photon acts as a herald for the second.

Second photon available for Hanbury-Brown and Twiss measurement or interference measurement.

Found g(2)(0) ~ 0 (single photons) andfringe visibility = 98%

Two-photon interference - Hong, Ou and Mandel 1987

“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac

Two photons in overlapping modes

50/50 beam splitter R=T=1/2

P = R T 2 = 1/2

Boson “clumping”. If one photon is present then it is easier to add a second.

Two-photon interference - Hong, Ou and Mandel 1987

“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac

Two photons in overlapping modes

50/50 beam splitter R=T=1/2

P = R T 2 = 1/2

Boson “clumping”. If one photon is present then it is easier to add a second.

Two-photon interference - Hong, Ou and Mandel 1987

“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac

Two photons in overlapping modes

50/50 beam splitter R=T=1/2

P = 0 !!!

Destructive quantum interference between the amplitudes for two reflections and two transmissions.

1. A bit about photons

2. Optical polarisation

3. Generalised measurements

4. State discriminationMinimum errorUnambiguousMaximum confidence

Maxwell’s equations in an isotropic dielectric medium take the form:

tc

t

EB

BE

BE

2

00

E, B and k are mutually orthogonal

For plane waves (and lab. beams that are not too tightly focussed) this means that the E and B fields are constrained to lie in the plane perpendicular to the direction of propagation.

k

E

B

BES 10

Consider a plane EM wave of the form

)(exp

)(exp

0

0

tkzitkzi

BBEE

If E0 and B0 are constant and real then the wave is said to be linearly polarised.

E

B

E

B

Polarisation is defined by an axis rather than by a direction:

If the electric field for the plane wave can be written in the form

)(exp)(0 tkziiE jiE

Then the wave is said to be circularly polarised.

B

E

For right-circular polarisation, an observer would see the fields rotating clockwise as the light approached.

The Jones representation

We can write the x and y components of the complex electric field amplitude in the form of a column vector:

y

x

iy

ix

y

x

eE

eEEE

0

0

0

0

The size of the total field tells us nothing about the polarisation so we can conveniently normalise the vector:

Horizontal polarisation

01

Vertical polarisation

10

Left circular polarisation

i1

21

Right circular polarisation

i1

21

One advantage of this method is that it allows us to describe the effects of optical elements by matrix multiplication:

Linear polariser(oriented to horizontal):

451111

,901000

,00001

21

Quarter-wave plate(fast axis to horizontal):

451

1,90

001

,00

012

1

i

iii

Half-wave plate(fast axis horizontal or vertical):

1001

The effect of a sequence of n such elements is:

BA

dcba

dcba

BA

nn

nn

11

11

We refer to two polarisations as orthogonal if

01*2 EE

This has a simple and suggestive form when expressed in terms of the Jones vectors:

0

0

0

iftoorthogonalis

1

1†

2

2

1

1*2

*2

1*21

*2

2

2

1

1

BA

BA

BABA

BBAA

BA

BA

There is a clear and simple mathematical analogy between the Jones vectors and our description of a qubit.

Poincaré Sphere

Optical polarization

Bloch Sphere

Electron spin

Spin and polarisation QubitsPoincaré and Bloch Spheres

Two state quantum system

We can realise a qubit as the state of single-photon polarisation

Horizontal

Vertical

Diagonal up

Diagonal down

Left circular

Right circular

0

102

1

102

1

102

1 i

102

1 i

1

1. A bit about photons

2. Optical polarisation

3. Generalised measurements

4. State discriminationMinimum errorUnambiguousMaximum confidence

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

mm

mm

pxpxdpdx

xxipxxdxpx

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.

1. A bit about photons

2. Optical polarisation

3. Generalised measurements

4. State discriminationMinimum errorUnambiguousMaximum confidence

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/ 2

0

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.

Maximum confidence measurements seek to maximise theconditional probabilities

for each state. For unambiguous discrimination these are all 1.

Bayes’ theorem tells us that

so the largest values of those give us maximum confidence.

)|( iiP

k kikk

iiii

i

iiiii p

pP

PpP

ˆˆ

)()|()|(

The solution we find is

where

11 ˆˆˆˆ ji

iii

jjj

p

ˆˆ

ˆ

Croke et al Phys. Rev. Lett. 96, 070401 (2006)

Example

• 3 states in a 2-dimensional space

• Maximum Confidence Measurement:

• Inconclusive outcome needed

Optimum probabilities• Probability of correctly

determining state maximised for minimum error measurement

• Probability that result obtained is correct maximised by maximum confidence measurement:

Results:

Minimum error

Maximum confidence

• Photons have played a central role in the development of quantum theory and the quantum theory of light continues to provide surprises.

• True single photons are hard to make but are, perhaps, the ideal carriers of quantum information.

• It is now possible to demonstrate a variety of measurement strategies which realise optimised POMs

• The subject of quantum optics also embraces atoms, ions molecules and solids ...

Conclusions