DEX-SMI Workshop on Quantum Statistical Inference, NII, 3 March 2009

Christoffer WittmannKatiuscia N. Cassemiro *1

Gerd LeuchsUlrik L. Andersen *2

Experimental implementation of near-optimal quantum measurements of optical coherent states

Masahiro TakeokaKenji TsujinoMasahide Sasaki

*1*2Daiji Fukuda

Go Fujii *3

Shuichiro Inoue *3 *3

Quantum optics: experimentally feasible approach to demonstrate quantum state discriminations

polarization (& location) encoding in single-photon states

Minimum error discrimination

encoding in coherent states

Huttner et al., Phys. Rev. A 54, 3783 (1996)

Unambiguous state discriminationClarke et al., Phys. Rev. A 63, 040305(R) (2001)

Collective measurements Fujiwara et al., Phys. Rev. Lett. 90, 167906 (2003)Pryde et al., Phys. Rev. Lett. 94, 220406 (2005)

Programmable unambiguous state discriminatorBartuskova et al., Phys. Rev. A 77, 034406 (2008)

etc.....

For applications?

Original motivation for the state discrimination

C. W. Helstrom 1976

Quantum noise in optical coherent states

Sender

0 10 1

00

Receiver

Non-orthogonality

Quantum Noise

for

5

SILEX

ETS-VIOICETS

TerraSAR-XDigitalcoherent

NeLS

1

10

100

1000

10000

1990 1995 2000 2005 2010 2015Launch year

Sen

sitiv

ity@

BER

=10-6

[Pho

tons

/bit]

Space qualified & planGround test

Trends of optical receiver sensitivity

Homodyne coherent PSK theoretical limit

IMDD

CoherentChallenge to quantum limit

Discrimination of binary coherent states

Min. error discrimination

Minimum Error Probability:

→ Projection onto the superpositions of coherent states

Binary Coherent States:

POVMBPSKcoherent

statesMeasurement

1010−−99

1010−−66

1010−−33

00 22 44 66 88 10 10

Photon number/pulsePhoton number/pulse

Bit

erro

r rat

eB

it er

ror r

ate

11

Minimum errorMinimum error((HelstromHelstrom bound)bound)

Quantum receivers

Homodyne limitHomodyne limit(Coherent optical communication)(Coherent optical communication)

R. S. Kennedy, RLE, MIT, QPR, R. S. Kennedy, RLE, MIT, QPR, 108, 219 (1973)108, 219 (1973)

Near optimal receiverNear optimal receiver（（Kennedy receiverKennedy receiver））

Coherent local oscillatorCoherent local oscillatorPhoton counterPhoton counter

S. J. S. J. DolinarDolinar, RLE, MIT, QPR, 111, 115, (1973), RLE, MIT, QPR, 111, 115, (1973)

Optimal receiver Optimal receiver （（DolinarDolinar receiverreceiver））

Coherent local oscillatorCoherent local oscillatorPhoton counterPhoton counterClassical feedbackClassical feedback

(infinitely fast!)(infinitely fast!)

No experiments have No experiments have beaten the homodyne limit!beaten the homodyne limit!

Contents

2. Practical near-optimal quantum receiver (Improvement of the Kennedy receiver)

2-1 Proposal and proof-of-principle experiment

1. Homodyne measurement

The optimal strategy within Gaussian operations and classical communication

Toward beating the homodyne limit:

2-2 Device: superconducting photon detector (TES)2-3 Theory: performance evaluation via the cut-off rate

1010−−99

1010−−66

1010−−33

00 22 44 66 88 10 10

Photon number/pulsePhoton number/pulse

Bit

erro

r rat

eB

it er

ror r

ate

11

Minimum errorMinimum error((HelstromHelstrom bound)bound)

Quantum receivers

Homodyne limit (SNL) Homodyne limit (SNL) (Coherent optical communication)(Coherent optical communication)

R. S. Kennedy, RLE, MIT, QPR, R. S. Kennedy, RLE, MIT, QPR, 108, 219 (1973)108, 219 (1973)

Kennedy receiverKennedy receiver

S. J. S. J. DolinarDolinar, RLE, MIT, QPR, 111, 115, (1973), RLE, MIT, QPR, 111, 115, (1973)

DolinarDolinar receiverreceiver

Best strategy within Best strategy within Gaussian operations and Gaussian operations and classical communication classical communication (feedback)(feedback)

Gaussian operations and classical communication(GOCC)

Gaussian operation

Classical communication

Gaussian operation

Eisert, et al, PRL 89, 137903 (2002)Fiurasek, PRL 89, 137904 (2002)Giedke and Cirac, PRA 66, 032316 (2002)

If is a Gaussian state, any classical communication does not help the protocol!

(for any trace decreasing Gaussian CP map, one can construct a corresponding trace preserving GCP map)

However, the receiver does not know which signal is coming..

Measurement via GOCC

Gaussian operations and classical communication(GOCC)

In our problem, and are Gaussian.

Does classical communication increase the distinguishability?

non-Gaussian state!

without CC

Gaussian measurement

Discrimination via Gaussian measurement without CC.

Optimal measurement under Bayesian strategy…

Average error probability

Homodyne measurement with

(independent on )

?

Gaussian unitary

operation M-mode G-measurements(without CC)

: measurement outcome

Input

Ancillae

G-meas.(without CC)

(N-M)-modeconditional state

Classical communication (conditional dynamics)

: pure Gaussian states

Homodyne measurement

measurement-dependentmeasurement-dependent

Classical communication does not increase the distinguishability

Homodyne limit

Minimum error discrimination of binary coherent states under Gaussian operation and classical communication is achieved by a simple homodyne detection

Limit of Gaussian operations

For multiple coherent states?multi-partite signals?

Takeoka and Sasaki, Phys. Rev. A 78, 022320 (2008)

Classical-quantum capacity with restricted (GOCC) measurement?

Contents

2. Practical near-optimal quantum receiver (Improvement of the Kennedy receiver)

2-1 Proposal and proof-of-principle experiment

1. Homodyne measurement

The optimal strategy within Gaussian operations and classical communication

Toward beating the homodyne limit:

2-2 Device: superconducting photon detector (TES)2-3 Theory: performance evaluation via the cut-off rate

Kennedy, RLE, MIT, QPR 108, 219 (1973)

discrim. error

On/off detection

Kennedy receiver

Local oscillator

BS

Displacement operation

Transmittance: T~1Photon detection

1010−−99

1010−−66

1010−−33

00 22 44 66 88 10 10

Photon number/pulsePhoton number/pulseB

it er

ror r

ate

Bit

erro

r rat

e

11Kennedy receiverKennedy receiver

Homodyne limitHomodyne limit

Local oscillator

Photon detection Input signals

001 1

T → 1

BS

0 photons

non-zero photons

001 1

Interference visibility

quantum efficiency, dark counts

Practical imperfections

0 2 4 6 8 10

-8

-6

-4

-2

0

Ave

rag

e er

ror

pro

bab

ility

L

og

10P

e

Average photon number

Homodyne Kennedy (ξ=0.99) Kennedy (ξ=0.9999) Kennedy (ξ=0.999999) Kennedy (ξ=0.99999999) Kennedy (ξ=0.9999999999)

n < 1

Visibility

0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.1

1

Ave

rag

e er

ror

pro

bab

ility

Average signal photon number

Homodyne limitHomodyne limitKennedy receiverKennedy receiver

(ideal)(ideal)

Helstrombound

Kennedy receiver at extremely weak signals

on/off detector

Kennedy receiverKennedy receiverdisplacement

Generalizing of the Kennedy receiver

on/off detector

Optimal DisplacementOptimal Displacement optimized γ

Squeezing + Displacement Squeezing + Displacement (Gaussian unitary operation)(Gaussian unitary operation)

on/off detector

optimized ζ and β

squeezer

Takeoka and Sasaki, Phys. Rev. A 78, 022320 (2008)

Average error probabilities

0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.1

1

Ave

rag

e er

ror

pro

bab

ility

Average signal photon number

KennedyKennedy

HomodyneHomodyne

HelstromHelstrom boundbound

DD((γγ ))

DD((ββ )+)+SS((ζζ ))

AO

Sig

Proof-of-principle experimentWittmann, et al., Phys. Rev. Lett. 101, 210501 (2008)

on/off detector

Optimal Displacement Optimal Displacement ReceiverReceiver

Average error probability (experimental)

*Detection efficiency compensated

* *

““ProofProof--ofof--principleprinciple”” demonstration succeeded!demonstration succeeded!

Wittmann, et al., Phys. Rev. Lett. 101, 210501 (2008)

Contents

2. Practical near-optimal quantum receiver (Improvement of the Kennedy receiver)

2-1 Proposal and proof-of-principle experiment

1. Homodyne measurement

The optimal strategy within Gaussian operations and classical communication

Toward beating the homodyne limit:

2-2 Device: superconducting photon detector (TES)2-3 Theory: performance evaluation via the cut-off rate

0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.1

1

BER

Average photon number

KennedyKennedy

Homodyne limitHomodyne limit

HelstromHelstromboundbound

Detector requirements

to beat the homodyne limit…

QE > 90%DC < 10-3

Detector

Visibility

ξ > 0.995

Advanced detectors?

Opt. disp. receiverOpt. disp. receiver

Transition Edge Sensor (TES)

TES: calorimetric detection of photonsFukuda et al., (2009) @AIST

@850nm

Contents

2. Practical near-optimal quantum receiver (Improvement of the Kennedy receiver)

2-1 Proposal and proof-of-principle experiment

1. Homodyne measurement

The optimal strategy within Gaussian operations and classical communication

Toward beating the homodyne limit:

2-2 Device: superconducting photon detector (TES)2-3 Theory: performance evaluation via the cut-off rate

Cut-off rate evaluation

3. Receiver implementation & simulation

1. (Classical) reliability function and cut-off rate

2. Quantum measurement attaining the maximum cut-off rate

4. Conclusions

Reliability function

R=k/N

0 0 1 0

N

N-kk

1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 0

message

code word

channel

0 0 1 0

error correctiontransmission rate:

detection

Average error probability (BER)

Detection error(w/o coding)

Mutual information (Shannon information) I(X:Y) Maximum R preserving

Pav

at

Reliability function Error bound for finite N

Reliability function and cut-off rate

R=k/N

0 0 1 0

N

N-kk

1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 0

message

code word

channel

0 0 1 0

message

error correctiontransmission rate:

Reliability function

detection

GallagerGallager, , Information Theory and Reliable CommunicationsInformation Theory and Reliable Communications, (1968)., (1968).

Reliability function and cut-off rate (classical)

: cut-off rate

Binary symmetric channel

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

E(R

)

R

: mutual information

Binary communication

sender receiver

homodyne or quantum receivers

BPSK coherent signal

- fixed single-shot measurement(non-adaptive, not collective)

Bendjaballah and Charbit, IEEE Trans. Info. Theory, 35, 1114 (1989).

Cut-off rate upper bound

Optimal quantum measurement strategies

- Minimum (average) error discrimination

- Unanimous voting discrimination

- Unambiguous state discrimination

We found that the following three strategies simultaneously attaining the upper bound of the cut-off rate;

Minimum (average) error discrimination

BPSKcoherent

statesMeasurement

Minimum Error Probability:

→ Projection onto the superpositions of coherent states

Cut-off rate:

I(X:Y) is also maximized.

Implementation: realtime adaptive feedbackDolinar receiver

S. J. S. J. DolinarDolinar, RLE, MIT, QPR, 111, 115, (1973), RLE, MIT, QPR, 111, 115, (1973)

Coherent state local oscillatorCoherent state local oscillator

Photon counterPhoton counter

Classical Classical (electrical)(electrical)feedbackfeedback

LOLO

or

Concept demonstration

Cook, Martin, and Geremia, Nature 446, 774 (2007)

Difficult to implement with high visibility & high QE detectors?

Unanimous voting discrimination

discrim. error

Cut-off rate: max.! discrim. error

On/off detection

Local oscillator

BS

Displacement operation

Transmittance: T~1Photon detection

Kennedy receiver

inconclusive result

Unambiguous state discriminationIvanovic, Phys. Lett. A 123, 257 (1987)Dieks, Phys. Lett. A 126, 303 (1988)Peres, Phys. Lett. A 128, 19 (1988)

Cut-off rate: Maximum!

inconclusive

Implementation of USD

signal

(a) click, (b) no

van Enk, Phys. Rev. A 66, 042313 (2002).

ancilla

(a) (b)

(a) no, (b) click

(a) no, (b) no

Signal decision

(a) click, (b) click inconclusivein practice,

inconclusive result

Intermediate between unambiguous & min. error discrimination

Optimal intermediate measurement

Chefles and Barnett, J. Mod. Opt. 45, 1295 (1998).

Reliability functions (& cut-off rate)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.1

0.2

0.3

0.4

0.5

E(R

)

R

Helstrom Kennedy Unambiguous Homodyne

BPSK coherent signal

Min. errorUnanimousUnambiguousHomodyne

Unambiguous

Minimum Error

Unanimous Voting

Local oscillator

Photon detection Input signals

001 1

T ~ 1

BS

0 photons

non-zero photons

001 1

interference visibility (mode matching)

detection efficiency, dark counts

Against the imperfections…

: mode match (visibility)

: quantum efficiency: dark counts

0 0.2 0.4 0.6 0.8 1photon number

0.10.20.30.40.5

tuc-

ffoetar

Ideal Kennedy (solid)

Homodyne limit (solid)

Kennedy

Cut-off rate performance (Kennedy receiver)

Cut-off rate performance

0 0.2 0.4 0.6 0.8 1 1.2 1.4photon number

-0.04

-0.02

0

0.02

0.04

tuc-

ffoetar

HecnereffidL

Kennedy receiver

ideal

Difference:

cut-

off r

ate HD

0 0.2 0.4 0.6 0.8 1 1.2 1.4photon number

-0.04

-0.020

0.020.04

tuc-

ffoetar

HecnereffidL

cut-

off r

ate

ideal

USD receiver

Optimal displacement receiver

Local oscillator

BS

Optimization taking into account practical imperfections

Transmittance: T~1

0 0.2 0.4 0.6 0.8 1photon number

-0.04

-0.02

0

0.02

0.04

tuc-

ffoetar

ODR

Kennedy

cut-

off r

ate

Comparable to the USD receiverfor n < 0.5

n

Easier to implement!

would be the first experimental demonstration beating the homodyne limit

Under construction…

Optimal displacement receiver with a TES

Conclusions

2. Near-optimal quantum receiver beyond the homodyne limit

1. Homodyne measurement is the optimal GOCC measurement for the minimum error discrimination of binary coherent states.

State discrimination via Gaussian operations and classical communication

Optimal displacement measurement

Proof-of-principle experiment

Figure of merits: - min. error probability- reliability function & cut-off rate

Simplest and robust scheme

Experiment beyond a “proof-of-principle” …

QE > 90%Detector

QE > 80%Detector

Mode matchξ > 0.990

Mode matchξ > 0.995

DC < 10-3

DC < 10-3