Download Quantum Metrology for Dynamical Systems

Quantum Metrology for Dynamical Systems
∗
Mankei Tsang
Department of Electrical and Computer Engineering
Department of Physics
National University of Singapore
[email protected]
http://mankei.tsang.googlepages.com/
Feb 27, 2014
∗
This material is based on work supported by the Singapore National Research Foundation Fellowship (NRF-NRFF2011-07).
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Quantum Limits to Classical Force Detection
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Caves et al., “On the measurement of a weak classical force coupled to a quantum-mechanical
oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341 (1980).
Yuen, “Contractive States and the Standard Quantum Limit for Monitoring Free-Mass
Positions,” Phys. Rev. Lett., 51, 719-722 (1983).
Caves, “Defense of the standard quantum limit for free-mass position,” Phys. Rev. Lett., 54,
2465-2468 (1985).
Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev.
Lett., 60, 385-388 (1988).
Braginsky and Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992).
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Quantum Optomechanics
Brooks et al., Nature 488, 476 (2012)
Safavi-Naeini et al., Nature 500, 185
(2013)
Purdy et al., PRX 3, 031012 (2013)
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Detection is Binary Hypothesis Testing
LIGO, Hanford
Rugar et al., Nature 430, 329 (2004).
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Statistical Binary Hypothesis Testing
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signal present: P (Y |H1 ), signal absent: P (Y |H0 ).
Type-I error probability (false-alarm probability):
P10 = Pr H̃(Y ) = H1 H0
Type-II error probability (miss probability):
P01
■
Average error probability:
= Pr H̃(Y ) = H0 H1
Pe = P10 P0 + P01 P1
■
(1)
(2)
(3)
Figure of merit: error exponent − ln Pe , usually − ln Pe ∝ Signal-to-Noise Ratio
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Quantum Detection Bounds
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C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic Press, New York,
1976).
Quantum hypothesis testing:
Pr(Y |H0 ) = tr[E(Y )ρ0 ],
■
p
1
1
P0 P1 F ≥ min Pe = (1 − ||P1 ρ1 − P0 ρ0 ||1 ) ≥
1 − 1 − 4P0 P1 F ≥ P0 P1 F.
E(Y )
2
2
√
■
(4)
Quantum bounds (Helstrom, Fuchs/van de Graaf):
p
■
Pr(Y |H1 ) = tr[E(Y )ρ1 ].
A† A,
||A||1 ≡ tr
For pure states,
p√
√
ρ1 ρ0 ρ1
2
.
||P1 ρ1 − P0 ρ0 ||1 =
p
1 − 4P0 P1 F ,
F ≡ tr
F = |hψ0 |ψ1 i|2 ,
(5)
(6)
and there exists a POVM such that the fidelity bound is attained.
Error exponent − ln PeHelstrom ≈ − ln P0 − ln P1 − ln F , fidelity exponent − ln F is a figure of
merit.
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Fidelity and Quantum Fisher Information
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■
Let F (x, x′ ) = |hψx |ψx′ i|2 .
Near F = 1,
1X
F (x, x + ∆x) ≈ 1 −
Jjk (x)∆xj ∆xk .
4 j,k
■
J(x) is the quantum Fisher information, determines quantum Cramér-Rao bound
mean-square error covariance ≥ J −1 .
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■
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(7)
(8)
Once we know F (x, x′ ), getting J(x) is easy, but not vice versa.
For detection, we want Pe ≪ 1, F ≪ 1, J is irrelevant
Pure-state Helstrom bound is attainable, low F means achievable low Pe .
For estimation, QCRB may not be attainable, high J doesn’t necessarily mean low error
QCRB useful as no-go theorem
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Helstrom Bound for Waveform Detection
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How to apply Helstrom’s bound to waveform detection?
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Larger Hilbert space: Purification, deferred measurements:
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Unitary discrimination:
ρ0 = U0 |ψihψ|U0† ,
Z
1
dtH0 (t) ,
U0 = T exp
i~
ρ1 = U1 |ψihψ|U1† ,
Z
1
U1 = T exp
dtH1 (x(t), t) ,
i~
2
†
F = hψ|U0 U1 |ψi .
(9)
(10)
(11)
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Quantum Optics 101
■
Interaction picture:
1
U0† U1 = T exp
i~
■
Z
dtHI (t) ,
HI (t) ≡ U0† (t, t0 ) [H1 (t) − H0 (t)] U0 (t, t0 ).
Suppose H1 − H0 = −qx(t),
Z
2
1
dtx(t)qI (t) .
F = T exp
i~
■
■
(12)
Assume qI (t) has linear dynamics (H0 quadratic with respect to Z, a vector of canonical
coordinate operators),
Z
qI (t) = g(t, t0 )Z + dt′ g(t, t′ )J(t′ ),
Cascading displacement operators:
Z
i
i
1
T exp
dtx(t)qI (t) ≈ eiφ e− ~ dtx(tN )g(tN ,t0 )Z . . . e− ~ dtx(t1 )g(t1 ,t0 )Z
i~
Z
′
i
dtx(t)g(t, t0 ) Z
= eiφ exp −
~
(13)
(14)
(15)
(16)
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Analytic Expression
■
■
Characteristic function is Fourier transform of Wigner:
D
E Z
e−iκZ =
dzW (z) exp(−iκz).
Assume initial state (light, mechanical) is Gaussian, Fourier transform of W (z) is Gaussian:
1
F = exp − 2
~
■
■
(17)
Z
dt
Z
dt′ x(t) : ∆qI (t)∆qI (t′ ) : x(t′ ) .
(18)
Trade-off between detection accuracy and quantum position localization
In optomechanics,
p
dq
=
,
dt
m
dp
2
= −mωm
q + κa† a,
dt
γ
da
√
= − a − iω0 a + γAin ,
dt
2
(19)
h: ∆qI (t)∆qI (t′ ) :i is a function of input optical state and dynamics.
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Standard Quantum Limit
Phase Homodyne
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Russian Referee?
Tsang and Nair, “Fundamental quantum limits to waveform detection,” PRA 86, 042115 (2012).
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Ponderomotive Squeezing
Brooks et al., Nature
488, 476 (2012)
Safavi-Naeini et al., Nature 500, 185 (2013)
Purdy et al., PRX 3, 031012 (2013)
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Overcoming SQL
Positive-mass oscillator
and negative-mass oscillator
Dynamical Pairs
Conjugate Pairs
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Kimble et al., PRD 65, 022002 (2001)
Quantum Noise Cancellation: Tsang and Caves, PRL 105, 123601 (2010)
Quantum-Mechanics-Free Subsystem: Tsang and Caves, PRX 2, 031016 (2012)
Backaction evasion
Undo ponderomotive squeezing
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Homodyne is Suboptimal
■
■
Suppose input field is in coherent state, output is also coherent state after QNC.
Coherent-state discrimination:
QNC
QNC
H0 : |Aout [x(t) = 0]i,
■
■
H1 : |Aout [x(t)]i.
(20)
Figure of merit: error exponent − ln Pe
− ln PeHomodyne
1
Helstrom
− ln Pe
≈
2
(21)
Phase Homodyne,
No Backaction Noise
Optimal measurement can save half of optical
power
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Quantum Optimal Measurements
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Kennedy: null the field for H0 :
H0 : |0i,
■
H1 : |Aout [x(t)] − Aout [0]i,
(22)
P01 = |h0|Aout [x(t)] − Aout [0]i|2 = F.
(23)
Error probabilities:
P10 = |hN 6= 0|N = 0i|2 = 0,
Photon Counter
nonzero count: choose
zero count: choose
− ln PeKennedy ≈ − ln PeHelstrom − (− ln P0 )
■
(24)
Dolinar?
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Stochastic Waveform Detection
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What if we don’t know x(t) exactly?
Fidelity upper/lower bounds still hold:
F =
■
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2
Z
1
dtx(t)qI (t) dP [x(t)] T exp
i~
Coherent state under H0 , mixed state under H1 :
Z
H0 : |Aout [0]i,
H1 :
dP [x(t)]|Aout [x(t)]ihAout [x(t)]|.
(25)
(26)
Kennedy receiver still has near-optimal error exponent:
Z
H0 : |0i,
H1 :
dP [x(t)]|Aout [x(t)] − Aout [0]ihAout [x(t)] − Aout [0]|,
P10 = 0,
■
Z
P01 =
Z
dP [x(t)] |h0|Aout [x(t)] − Aout [0]i|2 = F.
M. Tsang and R. Nair, PRA 86, 042115 (2012).
(27)
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Proof-of-Concept Experimental Proposals
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Squeezed Input State
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Caves, PRD 23, 1693 (1981).
GEO 600: Nature Phys. 7, 962 (2011)
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LIGO Hanford: Nature Photon. 7, 613 (2013)
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Squeezed Input State
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Squeezing increases h: ∆q(t)∆q(t′ ) :i without increase in optical power
Phase Homodyne,
No Backaction Noise
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LIGO nowhere near SQL yet
Optimal detection for squeezed state?
Fundamental quantum limit with respect to optical power?
◆
◆
Optical loss: noisy quantum metrology
Heisenberg limit?
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Open Systems
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Larger Hilbert space: bounds assume everything can be measured.
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For open systems, bounds are still valid but not tight
p√
√ 2
ρ1 ρ0 ρ1 , hopeless to compute
Mixed states: ||P1 ρ1 − P0 ρ0 ||1 , F = tr
Modified purification:
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2
†
F = max hψ|U0 UB U1 |ψi
UB
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Choose UB to tighten lower bound.
Hard to find optimal UB , lower bound may not be achievable, useful as no-go only.
Escher et al., Nature Phys. 7, 406 (2011) (quantum Fisher information)
(28)
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Single-Mode Optical Phase Detection
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Pure state:
2
X
2
P (n) exp(inφ) .
F = |hψ| exp(inφ)|ψi| = (29)
n
■
■
Characteristic function (Fourier transform)
Quantum Fisher information for QCRB on phase estimation:
F ≈ 1 − h∆n2 iφ2 ,
■
J = 4 ∆n2 .
e.g. coherent state:
Fcoh
φ
= exp −hni4 sin2
,
2
− ln Fcoh ∝ hni
(shot-noise scaling)
(30)
(31)
|φ| ≪ 1:
Fcoh ≈ exp −hniφ
2
,
− ln Fcoh ≈ hniφ2
(32)
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Modified Purification
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Model loss as beam splitter:
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Order of BS and modulation doesn’t matter
i
h †
†
,
UAB ≡ exp iκ a b + ab
■
UA ≡ exp(ia† aφ).
Fidelity for the modified purification:
2
†
F = hψ|UAB UB UA UAB |ψi
■
(33)
(34)
Assume
UB ≡ exp(−ib† bθ),
θ free parameter.
(35)
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Quantum Optics 102
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Use Heisenberg picture:
h
†
UB UA UAB
UAB
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J− ≡ a b,
■
†
†
†
i
= exp iµa a + iνb b − γ(a b − ab ) ,
1 †
†
J3 ≡
b b−a a ,
2
†
J+ ≡ ab ,
1 †
†
J≡
b b+a a ,
2
h
Λ3 = cos
■
ξ
2
−
i λξ3
sin
†
ξ
2
i−2
,ξ≡
q
(38)
λ23 + 4λ+ λ− .
eiΛ− J− |0iB = eiΛ− a b |0iB = |0iB .
Final result (η = cos2 κ = quantum efficiency):
2
X
2
n
a† a
Pa (n)z ,
F = hψ|z
|ψi = n
■
(37)
[J+ , J− ] = 2J3 , [J3 , J± ] = ±J± .
SU(2) disentangling theorem:
3 iΛ− J−
,
exp(iλ+ J+ + iλ− J− + iλ3 J3 ) = eiΛ+ J+ ΛJ
3 e
■
(36)
where µ ≡ φ cos2 κ − θ sin2 κ, ν ≡ φ sin2 κ − θ cos2 κ, γ ≡ (φ + θ) sin κ cos κ.
Define SU(2) operators:
†
■
†
moment-generating function, z transform
z ≡ ηeiφ + (1 − η)e−iθ .
(39)
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Jensen’s Inequality
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If z is real and positive, we can use Jensen’s inequality:
X
Pa (n)z n = hz n i ≥ z hni ,
F ≥ z 2hni ≡ Fz ,
n
■
− ln F ≤ C hni ,
(40)
shot-noise scaling bound.
When is z = ηeiφ + (1 − η)e−iθ (θ free parameter) real and positive?
η<
1
: any φ.
2
η≥
1
1−η
: | sin φ| ≤
, cos φ > 0.
2
η
(41)
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Shot-Noise Scaling Bound
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Supose |φ| ≪ 1 and |φ| ≪ (1 − η)/η, fidelity bound is Gaussian:
η
Fz ≈ exp −
hniφ2 .
1−η
Compare this with coherent state:
Fcoh ≈ exp −hniφ
■
■
■
■
2
.
(43)
Quantum enhancement of fidelity exponent limited by η:
− ln F .
■
(42)
η
η
(− ln Fcoh ) =
hniφ2 .
1−η
1−η
(44)
Generalize to lossy optical phase waveform detection, optomechanical force detection (no
optical cavity), etc.
Effect of optical cavity/complicated dynamics?
Similar idea applies to QCRB for lossy waveform estimation
Tsang, “Quantum Metrology for Open Dynamical Systems,” NJP 15, 073005 (2013).
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Quantum Ziv-Zakai Bounds
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Parameter estimation: prior P (x), likelihood P (y|x)
Ziv-Zakai bound on MSE:
1
E [x − x̃(y)]2 ≥
2
Z
∞
dτ τ
0
Z
∞
dx [P (x) + P (x + τ )] Pe (x, x + τ )
(45)
−∞
Pe (x, x + τ ) = average error probability for binary hypothesis testing:
P (y|H0 ) = P (y|x),
■
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■
■
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P (y|H1 ) = P (y|x + τ ),
1
2
h
p
P0 =
P (x)
,
P (x) + P (x + τ )
1 − 1 − 4P0 P1 F (x, x + τ )
Quantum: Pe (x, x + τ ) ≥
prove Heisenberg limit
⊗ν
√
√
Rivas-Luis:
, super-Heisenberg J
1 − ǫ|0i + ǫ|ψi
P1 = 1 − P0 . (46)
i
Tsang, PRL 108, 230401 (2012).
Multiparameter extensions possible.
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Quantum Transition-Edge Detectors
■
Beyond initial state and measurements, dynamics can also enhance metrology
Unstable/chaotic dynamics is most promising.
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Loschmidt echo (Peres):
■
H1 = H0 + ∆H,
■
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■
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U0,1 = T exp −i
Z
dtH0,1 ,
2
†
F = hψ|U0 U1 |ψi
F ≪ 1: Signature of quantum chaos, quantum phase transitions
Useful for detection near quantum
phase transitions
P z z
x
Ising model (H0,1 = −J j σj σj+1 + g0,1 σj ): can’t beat shot-noise scaling
OPO, Dicke: near threshold,
p
− ln F (g, g + ∆g) ∝ ∆g
(47)
(48)
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■
Tsang, PRA 88, 021801(R) (2013).
Continuous Quantum Hypothesis Testing
■
Tsang, PRL 108, 170502 (2012).
■
Tsang, QMTR 1, 84-109 (2013)
◆
◆
◆
Begging the question: To demonstrate quantum behavior, you can’t assume just quantum
mechanics to process your data!
Right way: Hypothesis testing against classical model
The classical model should be as quantum as possible
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Bell/CHSH etc.
Dynamical systems: Wigner, quasi-probability
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Experimental Collaborations
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Hidehiro Yonezawa (now at UNSW@ADFA), Akira Furusawa at U. Tokyo
◆
Iwasawa et al., PRL 111, 163602 (2013)
Position MSE
−16 2
(10
m )
1 (i)
(b)
Signal
RF source
Ti:S
to mixer
probe
AOM
OPO
AOM
EOM
SHG
2
LO
LO
Computer
Estimator
EOM
Oscilloscope
Estimate
Estimate
■
■
1.2
1
0.8
(i)
(p)
(ii)
(iii)
0.6 (iv)
0.4
(f )
(i)
(ii)
(iii)
0.005 (iv)
0.01
0
6
7
10
10
2
|α| (1/sec)
cavity-enhanced waveform QCRB
Warwick Bowen at U. Queensland
◆
◆
■
FPGA
(iv)
Trevor Wheatley, Elanor Huntington at UNSW@ADFA
◆
■
from RF
source
0.5 (iii)
0
Momentum MSE
−12
2 2 2
(10
kg m /s )
Mirror
Motion
Force MSE (N )
(a)
(q)
(ii)
Ang et al., “Optomechanical parameter estimation,” NJP 15, 103028 (2013)
Rheology: Fractional Brownian motion estimation
Michele Heurs at MPI Gravitational Physics, Hannover: QNC
Aaron Danner at NUS: Microwave Photonics
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Quantum Smoothing
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Tsang, PRL 102, 250403 (2009); PRA 80, 033840 (2009); 81, 013824 (2010).
Experiments:
◆
◆
◆
■
Wheatley et al., PRL 104, 093601 (2010).
Yonezawa et al., Science 337, 1514 (2012).
Iwasawa et al., PRL 111, 163602 (2013).
Rediscovery: Gammelmark, Julsgaard, Molmer, PRL 111, 160401 (2013).
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References
■
Quantum Bounds
◆ Detection: Tsang and Nair, PRA 86, 042115 (2012).
◆ QCRB: Tsang, Wiseman, Caves, PRL, 106, 090401 (2011); Iwasawa et
al., PRL 111, 163602 (2013).
◆ QZZB: Tsang PRL 108, 230401 (2012).
◆ Open Systems: Tsang, NJP 15, 073005 (2013).
◆ Stellar Interferometry: Tsang, PRL 108, 230401 (2012).
■
Phase Homodyne,
No Backaction Noise
Quantum Dynamics Control
◆ Quantum Noise Cancellation: Tsang and Caves, PRL 105, 123601
(2010); PRX 2, 031016 (2012).
◆ Transition-Edge: Tsang, PRA 88, 021801(R) (2013).
■
Signal Processing
◆ Smoothing: Tsang, PRL 102, 250403 (2009); PRA 80, 033840 (2009);
◆
◆
◆
■
■
81, 013824 (2010).
Hypothesis Testing: Tsang, PRL 108, 170502 (2012); Tsang, QMTR
1, 84-109 (2013).
Mismatched Quantum Filtering: Tsang, arXiv:1310.0291.
Optomechanical Parameter Estimation: Ang et al., NJP 15, 103028
(2013).
http://mankei.tsang.googlepages.com/
[email protected]
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