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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/

Oct 2016

∗This work is supported by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2011-07 and an MOETier 1 grant.

Summary

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(a)

(b)

M. Tsang, R. Nair, and Lu, Physical Review X 6, 031033 (2016).

Experiments

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■ Tang, Durak, and Ling (CQT Singapore),“Fault-tolerant and finite-error localization forpoint emitters within the diffraction limit,”Optics Express 24, 22004 (2016).

■ Yang, Taschilina, Moiseev, Simon, Lvovsky(Calgary/Russian Quantum Center/IFFSChengdu), “Far-field linear optical super-resolution via heterodyne detection in ahigher-order local oscillator mode,” e-printarXiv:1606.02662 (2016).

■ Tham, Ferretti, Steinberg (Toronto), “Beat-ing Rayleigh’s Curse by Imaging UsingPhase Information,” e-print arXiv:1606.02666(2016).

■ Paur, Stoklasa, Hradil, Sanchez-Soto, Re-hacek (Europe) “Achieving the ultimate op-tical resolution,” Optica 3, 1144 (2016).

Popular Coverage

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■ Viewpoint in APS Physics■ IoP Physics World and nanotechweb.org:

Seth Lloyd of the Massachusetts Instituteof Technology in the US is impressed. ‘This isawesome work and I am amazed that it hasn’tbeen done before,’ he says. ‘Perhaps everyonethought it was too good to be true.’

■ APS Physics Central■ Phys.org

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 ∼ λ

sinφ. (2)

■ Frieden (1966), Helstrom (1970), Lindegren(1978), Bobroff (1986), ...

■ 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).

Two Point Sources

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(a)

(b)

■ Λ(x) = 12

[

|ψ1(x)|2 + |ψ2(x)|2]

■ 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.■ Cramer-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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■ Cramer-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

100Cramer-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.■ Quantum state of light: density matrix (operator) ρ(θ) (positive-semidefinite), M temporal

modes: ρ⊗M

■ Born’s rule (generalized) P (Y |θ) = tr[E(Y )ρ⊗M (θ)], E(Y ) is called a positive operator-valuedmeasure (POVM).

■ Helstrom (1967): For any POVM

Σ ≥ J−1 ≥ K−1, (7)

Jµν =

∫

dY P (Y |θ)[

∂

∂θµlnP (Y |θ)

] [

∂

∂θνlnP (Y |θ)

]

, (8)

Kµν =M Re (trLµLνρ) , (9)

∂ρ

∂θµ=

1

2(Lµρ+ ρLµ) . (10)

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

■ Mixed states:

ρ =∑

n

Dn |en〉 〈en| , (11)

Lµ = 2∑

n,m;Dn+Dm 6=0

〈en| ∂ρ∂θµ

|em〉Dn +Dm

|en〉 〈em| . (12)

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|+ ǫ

2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2) 〈ψ1|ψ2〉 6= 0, (13)

|ψ1〉 =∫ ∞

−∞

dxψ(x−X1) |x〉 , |ψ2〉 =∫ ∞

−∞

dxψ(x−X2) |x〉 . (14)

■ 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

100Cramer-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

■ Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016)

∆θ22 ≥ 1

K22=

1

N∆k2. (15)

■ Nair and Tsang, e-print arXiv:1604.00937 (accepted by PRL): thermal sources witharbitrary 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 | , (16)

|φq〉 =∫ ∞

−∞

dxφq(x) |x〉 , (17)

φq(x) =

(

1

2πσ2

)1/4

Hq

(

x√2σ

)

exp

(

− x2

4σ2

)

. (18)

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

1

J (HG)22

=1

K22=

4σ2

N. (19)

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

Spatial-Mode Demultiplexing (SPADE)

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image plane

...

...

image plane

...

...

SLIVER

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Estimator

Image

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)

Theoretical Follow-up

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Dimensions Sources Theory Experimental proposals

1. Tsang, Nair, and Lu,Phys. Rev. X 6, 031033(2016)

1D Weak thermal (optical fre-quencies and above)

Quantum SPADE

2. Nair and Tsang, Optics Ex-press 24, 3684 (2016)

2D Thermal (any frequency) Semiclassical SLIVER

3. Tsang, Nair, and Lu,arXiv:1602.04655

1D Weak thermal, lasers Semiclassical SPADE, SLIVER

4. Nair and Tsang, accepted byPRL, arXiv:1604.00937

1D Thermal Quantum SLIVER

5. M. Tsang, arXiv:1605.03799 1D Weak thermal Quantum, Bayesian, Minimax SPADE

6. Ang, Nair, and Tsang,arXiv:1606.00603

2D Weak thermal Quantum SPADE, SLIVER

7. Tsang, arXiv:1608.03211 2D Weak thermal (multiplesources)

Quantum SPADE

8. Lu, Nair, Tsang,arXiv:1609.03025

2D Weak thermal Quantum, Bayesian, Chernoff SPADE, SLIVER

Other groups:

■ Lupo and Pirandola (York, UK), arXiv:1604.07367.■ Rehacek et al. (Europe), arXiv:1606.08332.■ Krovi, Guha, Shapiro (BBN/MIT), arXiv:1609.00684.

Moments of Arbitrary Source Distributions (Unpublished)

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3

iTEM

21

3

2

1

×104

0

2

1.5

1

0.5

|θ′10|

10−3 10−2 10−1

MSE

forθ′ 10

×10−5

2

4

6

8X moment

|θ′01|

10−3 10−2 10−1

MSE

forθ′ 01

×10−5

2

4

6

8Y moment

θ′20

10−2 10−1

MSE

forθ′ 20

10−6

10−5

10−4X2 moment

θ′02

10−2 10−1

MSE

forθ′ 02

10−6

10−5

10−4Y 2 moment

β(10, 01) ×10−3

5 10 15

MSE

forθ′ 11

10−6

10−5

XY moment

SPADE (simulated)

SPADE (theory)

direct imaging (simulated)

direct imaging (theory)

■ M. Tsang, arXiv:1608.03211, in preparation.

Experiments

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■ Tang, Durak, and Ling (CQT Singapore), “Fault-tolerant and finite-error localization for pointemitters within the diffraction limit,” Optics Express 24, 22004 (2016).

◆ SLIVER◆ Laser, classical noise

■ Yang, Taschilina, Moiseev, Simon, Lvovsky (Calgary/Russian Quantum Center/IFFS), “Far-fieldlinear optical superresolution via heterodyne 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, close to quantum limit

■ Paur, Stoklasa, Hradil, Sanchez-Soto, Rehacek (Europe), “Achieving the ultimate opticalresolution,” Optica 3, 1144 (2016).

◆ SPADE◆ laser, close to quantum limit

Quantum Computational Imaging

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■ Design quantum computer to

◆ Maximize information extraction◆ Reduce classical computational complexity

Quantum Technology 1.5

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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

100Cramer-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

image plane

...

...

Estimator

Image

Inversion

■ FAQ: https://sites.google.com/site/mankeitsang/news/rayleigh/faq■ email: 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.

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. (20)

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)

2D SPADE and SLIVER

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

Misalignment

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SPADE

photon-counting

array

image

plane

image

plane

object

plane

■ ξ ≡ |θ1 − 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

102Cramer-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