Experimental implementation of near-optimal quantum measurements of - - PowerPoint PPT Presentation

DEX-SMI Workshop on Quantum Statistical Inference, NII, 3 March 2009 Christoffer Wittmann Katiuscia N. Cassemiro *1 Gerd Leuchs Ulrik L. Andersen *2

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

Masahiro Takeoka Kenji Tsujino Masahide Sasaki

*1 *2

Daiji 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 discrimination

Clarke 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 discriminator

Bartuskova 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

1 1

Receiver

Non-orthogonality

Quantum Noise

for

5

SILEX ETS-VI OICETS TerraSAR-X Digital coherent NeLS 1 10 100 1000 10000 1990 1995 2000 2005 2010 2015 Launch year Sensitivity@BER=10-6 [Photons/bit] Space qualified & plan Ground test

Trends of optical receiver sensitivity

Homodyne coherent PSK theoretical limit I M D D

Coherent

Challenge to quantum limit

Discrimination of binary coherent states

• Min. error discrimination

Minimum Error Probability: → Projection onto the superpositions of coherent states Binary Coherent States: POVM BPSK coherent states Measurement

10 10−

−9 9

10 10−

−6 6

10 10−

−3 3

2 2 4 4 6 6 8 8 10 10 Photon number/pulse Photon number/pulse Bit error rate Bit error rate 1 1

Minimum error Minimum error ( (Helstrom Helstrom bound) bound)

Quantum receivers

Homodyne limit Homodyne 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 receiver Near optimal receiver （ （Kennedy receiver Kennedy receiver） ）

Coherent local oscillator Coherent local oscillator Photon counter Photon counter

• S. J.
• S. J. Dolinar

Dolinar, RLE, MIT, QPR, 111, 115, (1973) , RLE, MIT, QPR, 111, 115, (1973)

Optimal receiver Optimal receiver （ （Dolinar Dolinar receiver receiver） ）

Coherent local oscillator Coherent local oscillator Photon counter Photon counter Classical feedback Classical 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

10 10−

−9 9

10 10−

−6 6

10 10−

−3 3

2 2 4 4 6 6 8 8 10 10 Photon number/pulse Photon number/pulse Bit error rate Bit error rate 1 1

Minimum error Minimum error ( (Helstrom Helstrom 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 receiver Kennedy receiver

• S. J.
• S. J. Dolinar

Dolinar, RLE, MIT, QPR, 111, 115, (1973) , RLE, MIT, QPR, 111, 115, (1973)

Dolinar Dolinar receiver receiver

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

• peration

Classical communication Gaussian

• peration

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

• peration

M-mode G-measurements (without CC)

: measurement outcome

Input Ancillae G-meas. (without CC) (N-M)-mode conditional state

Classical communication (conditional dynamics)

: pure Gaussian states Homodyne measurement measurement-dependent measurement-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

• scillator

BS

Displacement operation Transmittance: T~1 Photon detection 10 10−

−9 9

10 10−

−6 6

10 10−

−3 3

2 2 4 4 6 6 8 8 10 10 Photon number/pulse Photon number/pulse Bit error rate Bit error rate 1 1 Kennedy receiver Kennedy receiver Homodyne limit Homodyne limit

Local

• scillator

Photon detection

Input signals

1 1

T → 1 BS 0 photons non-zero photons

1 1

Interference visibility

quantum efficiency, dark counts

Practical imperfections

2 4 6 8 10

• 8
• 6
• 4
• 2

Average error probability Log10Pe 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 Average error probability Average signal photon number

Homodyne limit Homodyne limit

Kennedy receiver Kennedy receiver (ideal) (ideal)

Helstrom bound

Kennedy receiver at extremely weak signals

• n/off detector

Kennedy receiver Kennedy receiver

displacement

Generalizing of the Kennedy receiver

• n/off detector

Optimal Displacement Optimal Displacement

• ptimized γ

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

• n/off detector
• ptimized 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 Average error probability Average signal photon number

Kennedy Kennedy Homodyne Homodyne Helstrom Helstrom bound bound D D( (γ γ ) ) D D( (β β )+ )+S S( (ζ ζ ) )

AO Sig

Proof-of-principle experiment

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

• n/off detector

Optimal Displacement Optimal Displacement Receiver Receiver

Average error probability (experimental)

*Detection efficiency compensated

* *

“ “Proof Proof-

• of
• f-
• principle

principle” ” 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

Kennedy Kennedy

Homodyne limit Homodyne limit Helstrom Helstrom bound bound

Detector requirements

to beat the homodyne limit…

QE > 90% DC < 10-3 Detector Visibility ξ > 0.995 Advanced detectors?

• Opt. disp. receiver
• Opt. disp. receiver

Transition Edge Sensor (TES)

TES: calorimetric detection of photons

Fukuda 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-k k

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 correction transmission 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-k k

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 correction transmission rate:

Reliability function

detection

Gallager Gallager, , Information Theory and Reliable Communications Information 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 0.6 0.0 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

BPSK coherent states Measurement Minimum Error Probability: → Projection onto the superpositions of coherent states Cut-off rate: I(X:Y) is also maximized.

Implementation: realtime adaptive feedback Dolinar receiver

• S. J.
• S. J. Dolinar

Dolinar, RLE, MIT, QPR, 111, 115, (1973) , RLE, MIT, QPR, 111, 115, (1973)

Coherent state local oscillator Coherent state local oscillator Photon counter Photon counter Classical Classical (electrical) (electrical) feedback feedback LO LO

• r

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

• scillator

BS

Displacement operation Transmittance: T~1 Photon detection

Kennedy receiver

inconclusive result

Unambiguous state discrimination

Ivanovic, 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 inconclusive in 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 0.7 0.0 0.1 0.2 0.3 0.4 0.5

E(R) R

Helstrom Kennedy Unambiguous Homodyne

BPSK coherent signal

• Min. error

Unanimous Unambiguous Homodyne

Unambiguous Minimum Error Unanimous Voting

Local

• scillator

Photon detection

Input signals

1 1

T ~ 1 BS 0 photons non-zero photons

1 1

interference visibility (mode matching) detection efficiency, dark counts

Against the imperfections…

: mode match (visibility) : quantum efficiency : dark counts

0.2 0.4 0.6 0.8 1 photon number 0.1 0.2 0.3 0.4 0.5

t u c - f f

• e

t a r

Ideal Kennedy (solid) Homodyne limit (solid) Kennedy

Cut-off rate performance (Kennedy receiver)

Cut-off rate performance

0.2 0.4 0.6 0.8 1 1.2 1.4 photon number

• 0.04
• 0.02

0.02 0.04

t u c - f f

• e

t a r

H

e c n e r e f f i d

L

Kennedy receiver

ideal

Difference:

cut-off rate

HD

0 0.2 0.4 0.6 0.8 1 1.2 1.4 photon number

• 0.04
• 0.02

0.02 0.04

t u c - f f

• e

t a r

H

e c n e r e f f i d

L

cut-off rate

ideal

USD receiver

Optimal displacement receiver

Local

• scillator

BS

Optimization taking into account practical imperfections

Transmittance: T~1

0.2 0.4 0.6 0.8 1 photon number

• 0.04
• 0.02

0.02 0.04

t u c - f f

• e

t a r

ODR Kennedy

cut-off rate

Comparable to the USD receiver for 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 match ξ > 0.990 Mode match ξ > 0.995 DC < 10-3 DC < 10-3