1 / 19

Killing Rayleigh’s Criterion by Quantum Measurements ∗

(arXiv:1511.00552)

Mankei Tsang, Ranjith Nair, and Xiaoming Lu

Department of Electrical and Computer EngineeringDepartment of Physics

National University of Singapore

mankei@nus.edu.sg

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

18 November 2015

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

Executive Summary

2 / 19

(a)

(b)

Imaging of One Point Source

3 / 19

Point-Spread Function of Hubble Space Telescope

4 / 19

Inferring Position of One Point Source

5 / 19

■ Given N detected photons on camera, root-mean-square error of estimating X1:

∆X1 =σ√N, σ ∼ λ

sinφ. (1)

■ Assume classical source (e.g., thermal source,laser source).

■ Derive from signal-to-noise ratio, classi-cal/quantum Cramér-Rao bounds, etc.

■ Known since Helstrom, Bobroff, etc.

Superresolution Microscopy

6 / 19

■ PALM, STED, STORM, etc.: Make fluorescentparticles radiate in isolation. Estimate their po-sitions accurately by locating the centroids.

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

(25:45)■ require special fluorescent particles (doesn’t

work for stars), slow■ e.g., Betzig, Optics Letters 20, 237 (1995).

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

Two Point Sources

7 / 19

(a)

(b)

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

Centroid and Separation Estimation

8 / 19

■ Bettens et al., Ultramicroscopy 77, 37 (1999);■ Ram, Ward, Ober, PNAS 103, 4457 (2006)■ Assume incoherent sources, Poissonian statis-

tics for CCD■ classical Cramér-Rao bound for centroid:

∆θ1 ∼σ√N, (2)

■ CRB for separation estimation: two regimes

◆ θ2 ≫ σ:

∆θ2 ∼2σ√N, (3)

◆ θ2 ≪ σ:

∆θ2 →2σ√N

×∞ (4)

Bound blows up when the two spots overlapsignificantly.

◆ Curse of Rayleigh’s criterion◆ Localization microscopy: must avoid over-

lapping spots.

(a)

(b)

Fisher Information and Cramér-Rao Bound

9 / 19

■ Cramér-Rao bounds:

∆θ1 ≥1

√

J (ipc)11∆θ2 ≥

1√

J (ipc)22(5)

J (ipc) is Fisher information for CCD (image-plane photon counting).■ Assume Gaussian PSF, similar behavior for other PSF■ Rayleigh’s curse: ∆θ2 blows up when Rayleigh’s criterion is violated.

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ

2)

0

1

2

3

4Quantum and classical Fisher information

K11

J(ipc)11

K22

J(ipc)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds

1/K22

1/J(ipc)22

Quantum Cramér-Rao Bounds

10 / 19

■ 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,

not magnification)

Σ ≥ J−1 ≥ K−1, (6)

Kµν =M trLµLν + LνLµ

2ρ, (7)

∂ρ

∂θµ=

1

2(Lµρ+ ρLµ) . (8)

■ Coherent sources: Tsang, Optica 2, 646 (2015).■ Mixed states [Braunstein and Caves, PRL 72, 3439 (1994)]:

ρ =∑

n

Dn |en〉 〈en| , (9)

Lµ = 2∑

n,m;Dn+Dm 6=0

〈en| ∂ρ∂θµ |em〉Dn +Dm

|en〉 〈em| . (10)

Quantum Optics for Thermal Sources

11 / 19

■ Mandel and Wolf, Optical Coherence and Quantum Optics; Goodman, Statistical Optics■ Consider a thermal source, e.g., stars, fluorescent particles.■ Coherence time ∼ 10 fs. Within each coherence 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|) , 〈ψ1|ψ2〉 6= 0, (11)

|ψ1〉 =∫ ∞

−∞

dxψ(x−X1) |x〉 , |ψ2〉 =∫ ∞

−∞

dxψ(x−X2) |x〉 , |x〉 = a†(x) |vac〉 . (12)

■ Multiphoton coincidence events are even rarer because ǫ≪ 1.■ Reproduces standard shot-noise model for photon counting (also used by Ram et al.).■ Sudarshan-Glauber■ Similar model for stellar interferometry in Tsang, PRL 107, 270402 (2011).

Classical and Quantum Cramér-Rao Bounds

12 / 19

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ2)

0

1

2

3

4Quantum and classical Fisher information

K11

J(ipc)11

K22

J(ipc)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds

1/K22

1/J(ipc)22

■ For any ψ(x) with constant phase,

∆θ2 ≥1√K22

=1

∆k√N. (13)

(For Gaussian ψ, σ = 1/(2∆k))■ Fujiwara JPA 39, 12489 (2006): there exists a POVM such that ∆θµ → 1/

√

Kµµ asymptoticallyfor large M .

Hermite-Gaussian basis

13 / 19

■ Consider POVM measuring the photon in Hermite-Gaussian basis:

E0 = |vac〉 〈vac| , (14)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.

1√

J (HG)22=

1√K22

=2σ√N. (18)

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

Spatial-Mode Demultiplexing (SPADE)

14 / 19

Binary SPADE

15 / 19

image plane

leaky modes

leaky modes

✸✷❂❁

✵ � ✹ ✻ ✽ ✶✵

❋✐s

❤❡r

✐♥❢♦

r♠❛

✁✐♦

♥✴

✭◆✂

✄☎

✆ ✮

✵✵✳�

✵✳✹✵✳✻

✵✳✽✶

❈❧✝✞✞✟❝✝❧ ✠✟✞✡☛☞ ✟✌✍✎☞✏✝✑✟✎✌

❏✒❍✓✔

✕✕ ✖ ❑✕✕

❏✒✗♣✘✔

✕✕

❏✒❜✔

✕✕

✸✷❂�

✵ ✶ ✁ ✂ ✹ ✺

❋✐s

❤❡r

✐♥❢♦

r♠❛✄

✐♦♥

✴✭✿

☎ ◆✆✝

❲☎ ✮

✵✵✳✁

✵✳✹✵✳✻

✵✳✽✶

✞✟✠✡☛☞ ✟✌✍✎☞✏✑✒✟✎✌ ✍✎☞ ✠✟✌❝ P❙✞❑✓✓

❏✔✕♣✖✗

✓✓❏

✔❜✗✓✓

Quantum Optics for Classical Sources

16 / 19

■ Classical thermal sources, linear optics, photon counting■ Compare with Shor’s algorithm, boson sampling, QKD, squeezed/NOON-state metrology, ...■ Why quantum optics/information? Lessons from Shannon

Computational Imaging

17 / 19

Conclusion

18 / 19

■ Rayleigh’s criterion is irrelevant.■ SPADE can achieve quantum bound via linear optics and

photon counting.■ Mankei Tsang, Ranjith Nair, and Xiaoming Lu, “Quantum

theory of superresolution for two incoherent optical pointsources,” e-print arXiv:1511.00552.

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

■ email: mankei@nus.edu.sg■ Nair and Tsang (under preparation): simpler measurement

schemes that work for any PSF and ǫ.

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds

1/K22

1/J(ipc)22

image plane

leaky modes

leaky modes

mankei@nus.edu.sg

Simulated Errors of Maximum-Likelihood Estimation

19 / 19✸✷❂❁

✵ ✵✳✺ ✶ ✶✳✺ �

▼❡❛

✁✲s

q✉

❛✂❡

❡✂✂

♦✂

✴✭✹

✄☎ ✆

✝✮

✵✵✳✺

✶✶✳✺

�

❙✐♠✞❧✟✠✡❞ ✡rr☛r☞ ❢☛r ❙P❆✌❊

✍❂❏✎✏❍●✑

✷✷ ✒ ✍❂❑✎✷✷

▲ ✒ ✍✓

▲ ✒ ✔✓

▲ ✒ ✍✓✓

✸✷❂❁

✵ ✵✳✺ ✶ ✶✳✺ �

▼❡❛

✁✲s

q✉

❛✂❡

❡✂✂♦

✂✴

✭✹✄

☎ ✆✝

✮✵

✵✳✺✶

✶✳✺�

❙✐♠✞❧✟✠✡❞ ✡rr☛r☞ ❢☛r ❜✐♥✟r② ❙P❆✌❊

✍❏✎✏✑✒

✓✓

▲▲

▲

Executive SummaryImaging of One Point SourcePoint-Spread Function of Hubble Space TelescopeInferring Position of One Point SourceSuperresolution MicroscopyTwo Point SourcesCentroid and Separation EstimationFisher Information and Cramér-Rao BoundQuantum Cramér-Rao BoundsQuantum Optics for Thermal SourcesClassical and Quantum Cramér-Rao BoundsHermite-Gaussian basisSpatial-Mode Demultiplexing (SPADE)Binary SPADEQuantum Optics for Classical SourcesComputational ImagingConclusionSimulated Errors of Maximum-Likelihood Estimation