Fundamental Quantum Limits toWaveform Sensing

Mankei Tsangeletmk@nus.edu.sg

http://mankei.tsang.googlepages.com/

Department of Electrical and Computer Engineering and Department of Physics

National University of Singapore

*Supported by Singapore National Research Foundation Fellowship (NRF-NRFF2011-07)

Fundamental Quantum Limits toWaveform Sensing – p. 1/23

Quantum Probability Theory

Wave:

Probability:

P (x) = |〈x|ψ〉|2. (1)

More than classical probability:

Fundamental Quantum Limits toWaveform Sensing – p. 2/23

Quantum Probability Experiments

Kippenberg and Vahala, Science 321, 1172(2008), and references therein.

LIGO, Nature Phys.7, 962 (2011).

Lee et al., Science 332, 330(2011).

Kimble, Nature 453, 1023 (2008).

Julsgaard, Kozhekin, andPolzik, Nature 413, 400 (2001).

Chou et al., Science 329, 1630(2010).

Castellanos-Beltran et al.,Nature Phys. 4, 928 (2008).

Neeley et al., Na-ture 467, 570 (2010)

O’Connell et al., Nature 464,697 (2010).

Fundamental Quantum Limits toWaveform Sensing – p. 3/23

Quantum Sensing/Metrology

Fundamental Limits: What is the ultimate sensitivity allowed by quantum mechanics?

Estimation: Optimize data processing

Control: Optimize experiment

Examples: optical interferometry, optical imaging, optomechanical force sensing(gravitational-wave detection), atomic magnetometry, electrometer, etc.

Fundamental Quantum Limits toWaveform Sensing – p. 4/23

Quantum Parameter Estimation

Quantum Cramér-Rao bound (QCRB):

E`

δx2´˙

∆h2¸

≥ ~2

4, δx ≡ x− x̃(y), ∂ρx

∂x=

1

2

“

hρx + ρxh†”

. (2)

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, NewYork, 1976); V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004);Nature Photon. 5, 222 (2011).

Beyond QCRB: M. Tsang, PRL 108, 230401 (2012); R. Nair, arXiv:1204.3761; seealso V. Giovannetti, S. Lloyd, L. Maccone, PRL 108, 260405 (2012); M. J. W. Hall et al.,PRA 85, 041802(R) (2012).

Fundamental Quantum Limits toWaveform Sensing – p. 5/23

Problem 1: x(t) Changes in Time

0 100 200 300 400 500 600 700 800 900 1000−80

−60

−40

−20

0

20

40

60

Realizations of Wiener Process

Optical phase, magnetic field, electric field, force, etc. can all vary randomly in time

Gravitational waves are stochastic [Buonanno et al., Phys. Rev. D 55, 3330 (1997);Apreda et al., Nuclear Physics B 631, 342 (2002)].

Stochastic processes characterized by a priori statistics (e.g. Langevin,Fokker-Planck, power spectral density)

Fundamental Quantum Limits toWaveform Sensing – p. 6/23

Problem 2: Continuous Dynamics and Measurements

Continuous coupling of time-varying signal to sensor

Continuous quantum dynamics of sensor

Continuous non-destructive measurements: must include measurement backaction

Braginsky and Khalili, Quantum Measurement (Cambridge University Press, 1992)

Fundamental Quantum Limits toWaveform Sensing – p. 7/23

Quantum Information Theory to the Rescue

Discretize time, and take continuous limit at the end

Measurements and dynamics described by a sequence of completely positive mapsrepresented by Kraus operators

P [y|x] =X

µN ,...,µ1

trh

K̂µN (yN |xN ) . . . K̂µ1 (y1|x1)ρ̂0K̂†µ1 (y1|x1) . . . K̂†µN

(yN |xN )i

Larger Hilbert space (Kraus representation theorem, principle of deferredmeasurements):

K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory (Springer, Berlin, 1983).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2010).

Fundamental Quantum Limits toWaveform Sensing – p. 8/23

Dynamical Quantum Cramér-Rao Bound

Fisher information matrix J(t, t′)

Define inverse of J(t, t′) asR

dt′J(t, t′)J−1(t′, τ) = δ(t− τ).

Z

dtdt′C(t, t′)˘

Eˆ

δx(t)δx(t′)˜

− J−1(t, t′)¯

≥ 0. (3)

Two components: J(t, t′) = J(Q)(t, t′) + J(C)(t, t′).

J(Q) is a two-time quantum covariance function:

J(Q)(t, t′) =4

~2EX

D

∆ĥ(t)∆ĥ(t′)E

, ĥ(t) ≡Z tJ

t0

dτÛ†(τ, t0)δĤ(τ)

δx(t)Û(τ, t0).

J(C) incorporates a priori waveform information

J(C)(t, t′) = EX

δ lnP [x]

δx(t)

δ lnP [x]

δx(t′)

ff

. (4)

M. Tsang, H. M. Wiseman, and C. M. Caves, PRL 106, 090401 (2011).

Fundamental Quantum Limits toWaveform Sensing – p. 9/23

Example 1: Adaptive Optical Phase Estimation

unpublished: φ(t) =R ∞

−∞dτh(t− τ)x(τ),

δx2QCRB =

Z ∞

−∞

dω

2π

1

4|h(ω)|2S∆Î(ω) + 1/Sx(ω), (5)

e.g. Scoh∆Î

(ω) =P̄

~ω0, SOUx (ω) =

κ

ω2 + ǫ2. (6)

Filtering: Berry and Wiseman, PRA 65, 043803 (2002); 73, 063824 (2006).

Filtering does not saturate QCRB!

Fundamental Quantum Limits toWaveform Sensing – p. 10/23

Classical Estimation

Filtering, Prediction: real-time or advanced estimation

Smoothing: delayed estimation, more accurate when x(t) is a stochastic process

Fundamental Quantum Limits toWaveform Sensing – p. 11/23

Smoothing for Quantum Sensing

M. Tsang, PRL 102, 250403 (2009).

Schrödinger picture (density operator, POVM)

Smoothing estimation of quantum observables makes sense only if they have classicalprobability representations, e.g., QND observables (Yanagisawa arXiv:0711.3885),nonnegative quasi-probability functions (Tsang, PRA 80, 033840 (2009); 81, 013824(2010)).

Temporal Nonclassicality: Leggett-Garg inequality, weak valuesFundamental Quantum Limits toWaveform Sensing – p. 12/23

Experimental Demonstration

Optical phase-locked loop with smoother: M. Tsang, J. H. Shapiro, and S. Lloyd, PRA78, 053820 (2008); 79, 053843 (2009); M. Tsang, PRA 80, 033840 (2009):

Wheatley et al., Phys. Rev. Lett. 104, 093601 (2010); Yonezawa et al., Science (2012);PRACQSYS poster

very close to QCRB for coherent state.

QCRB can be lowered by squeezing Fundamental Quantum Limits toWaveform Sensing – p. 13/23

Example 2: Optomechanical Force Estimation

δf2QCRB =

Z ∞

−∞

dω

2π

1

(4/~2)S∆q̂(ω) + 1/Sf (ω). (7)

Smoothing can’t saturate QCRB due to the presence of measurement backactionnoise

Standard Quantum Limit (Braginsky, Caves et al.): backaction noise cannot beremoved.

Fundamental Quantum Limits toWaveform Sensing – p. 14/23

Quantum Noise Cancellation (QNC)

Coherent Feedforward Quantum Control

M. Tsang and C. M. Caves, PRL 105, 123601 (2010).

Fundamental Quantum Limits toWaveform Sensing – p. 15/23

Noise Cancellation

Fundamental Quantum Limits toWaveform Sensing – p. 16/23

Quantum-Mechanics-Free Subsystems

Another way of thinking about QNC:

Positive-mass oscillator

back-action

force

external

force

Negative-mass oscillator

back-action

force

Dynamical Pairs

Conjugate Pairs

Positive-mass oscillator

and negative-mass oscillator

{Q,Π} are dynamical QND observables:

Q̇ = Π/m, Π̇ = −mω2Q. (8)

no measurement backaction.

M. Tsang and C. M. Caves, PRX 2, 031016 (2012): By engineering the Hamiltonian, itis possible to make a subsystem of observables

QNDobey any linear/nonlinear classical dynamics.

Fundamental Quantum Limits toWaveform Sensing – p. 17/23

Optimal Force Estimation

QNC + Smoothing saturate QCRB for coherent state.

Optical squeezing of input light can lower the QCRB.

Fundamental Quantum Limits toWaveform Sensing – p. 18/23

Example 3: Magnetometry

Linear Gaussian smoother for magnetometry: Petersen and Mølmer, PRA 74, 043802(2006).

QNC:Julsgaard, Kozhekin, and Polzik, Nature 413, 400 (2001).Wasilewski et al., PRL 104, 133601 (2010).

Fundamental Quantum Limits toWaveform Sensing – p. 19/23

Quantum Waveform Detection

Helstrom Bounds: Pe ≥ 12`

1 −√

1 − 4P0P1F´

.

Fidelity: F = EX |〈ψ|T expR

dt∆HI(t)|ψ〉|2,∆HI(t) ≡ U†0 (t, t0) [H1(t) −H0(t)]U0(t, t0).

Wigner remains Gaussian, H1 −H0 = −qx(t):

F = EX exp

»

−14

Z

dtdt′x(t)x(t′)J(t, t′)

–

, J(t, t′) =4

~2

˙

: ∆q(t)∆q(t′) :¸

. (9)

M. Tsang and R. Nair, arXiv:1204.3697.

Fundamental Quantum Limits toWaveform Sensing – p. 20/23

Computation of Likelihood Ratio

Quantum Duncan-Kailath formula for Gaussian measurements in terms of filteringestimates conditioned on either hypotheses µj :

Λ =P (Y |H1)P (Y |H0)

=Λ1

Λ0, Λj = exp

„Z

dy⊤R−1µj −1

2

Z

dtµ⊤j R−1µj

«

. (10)

Quantum Snyder’s formula for Poisson measurements:Λj = exp

`R

dy lnµj −R

dtµj´

.

M -ary hypothesis testing: P (j|Y ) = ΛjPjPj ΛjPj

.

M. Tsang, PRL 108, 170502 (2012).

Fundamental Quantum Limits toWaveform Sensing – p. 21/23

Quantum Receiver Design

Homodyne is sub-optimal

Kennedy receiver is near-optimal (optimal error exponent), even if x(t) is stochastic.

OPABS

displace by

Photon detector

M. Tsang and R. Nair, arXiv:1204.3697.

Fundamental Quantum Limits toWaveform Sensing – p. 22/23

Outlook

Waveform EstimationFundamental Limits: M. Tsang, H. M. Wiseman, and C. M. Caves, PRL 106, 090401 (2011).

Estimation: M. Tsang, PRL 102, 250403 (2009); PRA 80, 033840 (2009); 81, 013824 (2010).

Control: M. Tsang and C. M. Caves, PRL 105, 123601 (2010); PRX 2, 031016 (2012).

Waveform DetectionFundamental Limits/Control: M. Tsang and R. Nair, arXiv:1204.3697.

Estimation: M. Tsang, PRL 108, 170502 (2012).

Parameter Estimation Beyond CRBQuantum Ziv-Zakai Bound: M. Tsang, PRL 108, 230401 (2012).

Rate distortion: R. Nair, arXiv:1204.3761.

ImagingStellar Interferometry: M. Tsang, PRL 107, 270402 (2011).

Waveform Parameter Estimation

We are hiring postdocs/PhD students!

Supported by Singapore National Research FoundationFellowship (NRF-NRFF2011-07).

http://www.nrf.gov.sg/nrf/otherProgrammes.aspx?id=142

OPABS

displace by

Photon detector

Fundamental Quantum Limits toWaveform Sensing – p. 23/23

http://www.nrf.gov.sg/nrf/otherProgrammes.aspx?id=142

Quantum Probability TheoryQuantum Probability ExperimentsQuantum Sensing/MetrologyQuantum Parameter EstimationProblem 1: $x(t)$ Changes in TimeProblem 2: Continuous Dynamics and MeasurementsQuantum Information Theory to the RescueDynamical Quantum Cram'er-Rao BoundExample 1: Adaptive Optical Phase EstimationClassical EstimationSmoothing for Quantum SensingExperimental DemonstrationExample 2: Optomechanical Force EstimationQuantum Noise Cancellation (QNC)Noise CancellationQuantum-Mechanics-Free SubsystemsOptimal Force EstimationExample 3: MagnetometryQuantum Waveform DetectionComputation of Likelihood RatioQuantum Receiver DesignOutlook