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Quantum Metrology Kills Rayleigh’s Criterion ∗

Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, and Mankei Tsang

mankei@nus.edu.sg

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

June 2016

∗This work is supported by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2011-07.

Summary

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(a)

(b)

Imaging of One Point Source

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Point-Spread Function of Hubble Space Telescope

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Inferring Position of One Point Source

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■ Classical source■ Given N detected photons, mean-square error:

∆X21 =σ2

N, (1)

σ ∼ λsinφ

. (2)

■ Helstrom, Lindegren (astrometry), Bobroff, ...■ Full EM, full quantum: Tsang, Optica 2,

646 (2015).

Superresolution Microscopy

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■ PALM, STED, STORM, etc.: isolate emitters.Locate centroids.

■ https://www.youtube.com/watch?v=2R2ll9SRCeo

(25:45)■ Special fluorophores■ slow■ doesn’t work for stars■ e.g., Betzig, Optics Letters 20, 237

(1995).

■ For a review, see Moerner, PNAS 104, 12596(2007).

https://www.youtube.com/watch?v=2R2ll9SRCeo

Two Point Sources

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(a)

(b)

■ Rayleigh’s criterion (1879): requires θ2 &σ (heuristic)

Centroid and Separation Estimation

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■ Bettens et al., Ultramicroscopy 77,

37 (1999); Van Aert et al., J. Struct.

Biol. 138, 21 (2002); Ram, Ward, Ober,

PNAS 103, 4457 (2006).

■ Incoherent sources, Poisson statistics■ X1 = θ1 − θ2/2, X2 = θ1 + θ2/2.■ Cramér-Rao bound for centroid:

∆θ21 ≥σ2

N. (3)

■ CRB for separation estimation: two regimes

◆ θ2 ≫ σ:

∆θ22 ≥4σ2

N, (4)

◆ θ2 ≪ σ:

∆θ22 →4σ2

N×∞ (5)

◆ Rayleigh’s curse◆ PALM/STED/STORM: avoid Rayleigh

(a)

(b)

Rayleigh’s Curse

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■ Cramér-Rao bounds:

∆θ21 ≥1

J (direct)11∆θ22 ≥

1

J (direct)22(6)

J (direct) is Fisher information for CCD■ Gaussian PSF, similar behavior for other PSF

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ

2)

0

0.5

1

1.5

2

2.5

3

3.5

4Classical Fisher information

J(direct)11

J(direct)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bound on separation error

Direct imaging (1/J(direct)22 )

Quantum Information

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■ CCD is just one measurement method. Quantum mechanics allows infinite possibilities.■ Helstrom: For any measurement (POVM) of an optical state ρ⊗M (M here is number of copies)

Σ ≥ J−1 ≥ K−1, (7)

Kµν =M Re (trLµLνρ) , (8)∂ρ

∂θµ=

1

2(Lµρ+ ρLµ) . (9)

■ Ultimate amount of information in the photons■ Coherent sources: Tsang, Optica 2, 646 (2015).■ Mixed states:

ρ =∑

n

Dn |en〉 〈en| , (10)

Lµ = 2∑

n,m;Dn+Dm 6=0

〈en| ∂ρ∂θµ |em〉Dn +Dm

|en〉 〈em| . (11)

Quantum Optics

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■ Mandel and Wolf, Optical Coherence and Quantum Optics ; Goodman, Statistical

Optics

■ Thermal sources, e.g., stars, fluorescent particles.■ Coherence time ∼ 10 fs. Within each coher-

ence time interval, average photon number ǫ≪ 1 at optical frequencies (visible, UV, X-ray, etc.).

■ Quantum state at image plane:

ρ = (1− ǫ) |vac〉 〈vac|+ ǫ2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2) 〈ψ1|ψ2〉 6= 0, (12)

|ψ1〉 =∫ ∞

−∞

dxψ(x−X1) |x〉 , |ψ2〉 =∫ ∞

−∞

dxψ(x−X2) |x〉 . (13)

■ derive from zero-mean Gaussian P function■ Multiphoton coincidence: rare, little information as ǫ≪ 1 (homeopathy)■ Similar model for stellar interferometry in Gottesman, Jennewein, Croke, PRL 109, 070503

(2012); Tsang, PRL 107, 270402 (2011).

Plenty of Room at the Bottom

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θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ2)

0

1

2

3

4Quantum and classical Fisher information

K11

J(direct)11

K22

J(direct)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

■ Tsang, Nair, and Lu, e-print arXiv:1511.00552

∆θ22 ≥1

K22=

1

N∆k2. (14)

■ Nair and Tsang, e-print arXiv:1604.00937: thermal sources with arbitrary ǫ■ Hayashi ed., Asymptotic Theory of Quantum Statistical Inference ; Fujiwara JPA 39,

12489 (2006): there exists a POVM such that ∆θ2µ → 1/Kµµ, M → ∞.

Hermite-Gaussian Basis

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■ project the photon in Hermite-Gaussian basis:

E1(q) = |φq〉 〈φq | , (15)

|φq〉 =∫ ∞

−∞

dxφq(x) |x〉 , (16)

φq(x) =

(

1

2πσ2

)1/4

Hq

(

x√2σ

)

exp

(

− x2

4σ2

)

. (17)

■ Assume PSF ψ(x) is Gaussian (common).

1

J (HG)22=

1

K22=

4σ2

N. (18)

■ Maximum-likelihood estimator can saturate the classical bound asymptotically for large N .

Spatial-Mode Demultiplexing (SPADE)

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image plane

...

...

image plane

...

...

Binary SPADE

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image plane

image plane

leaky modes

leaky modes

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ

2)

0

0.2

0.4

0.6

0.8

1Classical Fisher information

J(HG)22 = K22

J(direct)22

J(b)22

θ2/W0 1 2 3 4 5

Fisher

inform

ation/(π

2N/3W

2)

0

0.2

0.4

0.6

0.8

1Fisher information for sinc PSF

K22

J(direct)22

J(b)22

Numerical Performance of Maximum-Likelihood Estimators

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θ2/σ0 0.5 1 1.5 2

Mean-squareerror/(4σ2/L

)

0

0.5

1

1.5

2Simulated errors for SPADE

1/J′(HG)22 = 1/K

′22

L = 10L = 20L = 100

θ2/σ0 0.5 1 1.5 2

Mean-squareerror/(4σ2/L

)

0

0.5

1

1.5

2Simulated errors for binary SPADE

1/J′(b)22

L = 10L = 20L = 100

■ L = number of detected photons■ biased, < 2×CRB.

Minimax/Bayesian

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Van Trees inequality for any biased/unbiased estimator (e-print arXiv:1605.03799)

Quantum/SPADE: supθ

Σ22(θ) ≥4σ2

N, Direct imaging: sup

θΣ

(direct)22 (θ) ≥

σ2√N. (19)

θ/σ0 0.2 0.4 0.6 0.8 1

MSE

/(4σ2/L)

0

0.5

1

1.5

2(a) SPADE with ML

CRBL = 100L = 200L = 500L = 1000

θ/σ0 0.2 0.4 0.6 0.8 1

MSE

/(4σ2/L)

0

0.5

1

1.5

2(b) SPADE with modified ML

CRBL = 100L = 200L = 500L = 1000

θ/σ0 0.2 0.4 0.6 0.8 1

MSE

/(4σ2/L)

0

5

10

15

20(c) Direct imaging with ML

CRBL = 100L = 200L = 500L = 1000

Photon Number L102 103

supθMSE

/σ2

10−2

10−1

(d) Worst-case error

Quantum/SPADE limitSPADE (ML)SPADE (modified ML)Direct-imaging limitDirect imaging (ML)

SLIVER

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E � � i m � � o �

I m � � e

Inversion

■ SuperLocalization via Image-inVERsion interferometry■ Nair and Tsang, Opt. Express 24, 3684 (2016).

■ Laser Focus World, Feb 2016 issue.■ Classical theory/experiment of image-inversion interferometer: Wicker, Heintzmann,

Opt. Express 15, 12206 (2007); Wicker, Sindbert, Heintzmann, Opt. Express 17,

15491 (2009)

2D SPADE and SLIVER

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Ang, Nair, Tsang, e-print arXiv:1606.00603

Misalignment

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photon-counting

array

image

plane

image

plane

object

plane

■ ξ ≡ |θ̌1 − θ|/σ ≪ 1■ Overhead photons N1 ∼ 1/ξ2■ ξ = 0.1, N1 ∼ 100.■ CRB for Xs = θ1 ± θ2/2

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ2)

0

0.2

0.4

0.6

0.8

1Fisher information for misaligned SPADE

J(HG)22 (ξ = 0)

J(HG)22 (ξ = 0.1)

J(HG)22 (ξ = 0.2)

J(HG)22 (ξ = 0.3)

J(HG)22 (ξ = 0.4)

J(HG)22 (ξ = 0.5)

J(direct)22

θ2/σ0 0.5 1 1.5 2

mean-square

error/(4σ2/L)

0

2

4

6

8

10

12Simulated errors for misaligned binary SPADE

1/J′(b)22 (ξ = 0.1)

1/J′(direct)22

L = 10L = 100L = 1000

10−1 100 101

Mean-square

error/(2σ2/N

)

100

101

102Cramér-Rao bounds on localization error

Hybrid (ξ = 0)Hybrid (ξ = 0.1)Hybrid (ξ = 0.2)Hybrid (ξ = 0.3)Hybrid (ξ = 0.4)Hybrid (ξ = 0.5)Direct imaging

Follow-up

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Dimensions Sources Theory Experimental pro-posals

1. Tsang, Nair,and Lu, e-printarXiv:1511.00552

1D Weak thermal(optical frequen-cies and above)

Quantum SPADE

2. Nair and Tsang,Optics Express24, 3684 (2016)

2D Thermal (any fre-quency)

Semiclassical SLIVER

3. Tsang, Nair,and Lu, e-printarXiv:1602.04655

1D Weak thermal,lasers

Semiclassical SPADE, SLIVER

4. Nair andTsang, e-printarXiv:1604.00937

1D Thermal Quantum SLIVER

5. Ang, Nair, andTsang, e-printarXiv:1606.00603

2D Weak thermal Quantum SPADE, SLIVER

■ Lupo and Pirandola, e-print arXiv:1604.07367.■ Bayesian/minimax: M. Tsang, e-print arXiv:1605.03799.■ More to come!

Experiments

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■ Tang, Durak, and Ling (CQT Singapore), “Fault-tolerant and finite-error localization for pointemitters within the diffraction limit,” e-print arXiv:1605.07297 (2016).

◆ SLIVER◆ Laser, classical noise

■ Yang, Taschilina, Moiseev, Simon, Lvovsky (Calgary), “Far-field linear optical superresolution viaheterodyne detection in a higher-order local oscillator mode,” e-print arXiv:1606.02662 (2016).

◆ Mode heterodyne◆ Laser

■ Tham, Ferretti, Steinberg (Toronto), “Beating Rayleigh’s Curse by Imaging Using PhaseInformation,” e-print arXiv:1606.02666 (2016).

◆ SPADE◆ single-photon sources, quantum

Quantum Computational Imaging

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■ Design quantum computer to

◆ Maximize information extraction◆ Reduce classical computational complexity

Quantum Metrology with Classical Sources

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■ Thermal/fluorescent/laser sources, linear optics, photon counting■ Compare with other QIP applications:

◆ QKD (Bennett et al.)◆ Nonclassical-state metrology (Yuen, Caves)◆ Shor’s algorithm◆ Quantum simulations (Feynman, Lloyd)◆ Boson sampling (Aaronson)

■ Microscopy (physics, chemistry, biology, engineering), telescopy (astronomy), radar/lidar(military), etc.

■ Classical sources are more robust to losses.■ Current microscopy limited by photon shot noise in EMCCD (see, e.g., Pawley ed., Handbook of

Biological Confocal Microscopy).■ Ground telescopes limited by atmospheric turbulence, space telescopes are

diffraction/shot-noise-limited.■ Linear optics/photon-counting technology is mature

Although any given scheme can be explained by a semiclassical model, quantum metrologyremains a powerful tool for exploring the ultimate performance.

Quantum Metrology Kills Rayleigh’s Criterion

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θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

image plane

...

...

Estimator

� � � � e

� ersion

■ FAQ: https://sites.google.com/site/mankeitsang/news/rayleigh/faq■ email: mankei@nus.edu.sg

mankei@nus.edu.sg

ǫ ≪ 1 Approximation

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■ Chap. 9, Goodman, Statistical Optics:

“If the count degeneracy parameter is much less than 1, it is highly probable that there will be either zero or

one counts in each separate coherence interval of the incident classical wave. In such a case the classical

intensity fluctuations have a negligible ”bunching” effect on the photo-events, for (with high probability) the

light is simply too weak to generate multiple events in a single coherence cell.

■ Zmuidzinas (https://pma.caltech.edu/content/jonas-zmuidzinas), JOSA A 20, 218 (2003):

“It is well established that the photon counts registered by the detectors in an optical instrument follow

statistically independent Poisson distributions, so that the fluctuations of the counts in different detectors are

uncorrelated. To be more precise, this situation holds for the case of thermal emission (from the source, the

atmosphere, the telescope, etc.) in which the mean photon occupation numbers of the modes incident on the

detectors are low, n ≪ 1. In the high occupancy limit, n ≫ 1, photon bunching becomes important in that it

changes the counting statistics and can introduce correlations among the detectors. We will discuss only the

first case, n ≪ 1, which applies to most astronomical observations at optical and infrared wavelengths.”

■ Hanbury Brown-Twiss (post-selects on two-photon coincidence): poor SNR, obsolete fordecades in astronomy.

■ See also Labeyrie et al., An Introduction to Optical Stellar Interferometry, etc.■ Fluorescent particles: Pawley ed., Handbook of Biological Confocal Microscopy, Ram, Ober,

Ward (2006), etc., may have antibunching, but Poisson model is fine because of ǫ ≪ 1.

SummaryImaging of One Point SourcePoint-Spread Function of Hubble Space TelescopeInferring Position of One Point SourceSuperresolution MicroscopyTwo Point SourcesCentroid and Separation EstimationRayleigh's CurseQuantum InformationQuantum OpticsPlenty of Room at the BottomHermite-Gaussian BasisSpatial-Mode Demultiplexing (SPADE)Binary SPADENumerical Performance of Maximum-Likelihood EstimatorsMinimax/BayesianSLIVER2D SPADE and SLIVERMisalignmentFollow-upExperimentsQuantum Computational ImagingQuantum Metrology with Classical SourcesQuantum Metrology Kills Rayleigh's Criterion1 Approximation