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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?
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