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QUANTUM METROLOGY�WITH A LOSSLESS MACH-ZEHNDER INTERFEROMETER �USING PHOTON-COUNTING DETECTION �FOR A SEQUENCE OF NON-ADAPTIVE, SEMI-ADAPTIVE �AND ADAPTIVE MEASUREMENTS

by Shreyas Sadugol

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OUTLINE

Introduction and Motivation
Mathematical framework (Bayesian Estimation Theory)
Physical framework
Non-Adaptive measurement Strategy
Semi-Adaptive measurement Strategy
Fully Adaptive measurement Strategy
Conclusions

References

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INTRODUCTION AND MOTIVATION

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INTRODUCTION AND MOTIVATION

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Input states

Interaction of input with system

Detection

Input states

Interaction of input with system

Detection

Input states

Interaction of input with system

Detection

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INTRODUCTION AND MOTIVATION

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Input states

Interaction of input with system

Detection

Input states

Interaction of input with system

Detection

Mach-Zehnder Interferometer

Photon-counting detection

Entangled input

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INTRODUCTION AND MOTIVATION

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Input states

Interaction of input with system

Detection

Input states

Interaction of input with system

Detection

Mach-Zehnder Interferometer

Light Detection

Light input

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INTRODUCTION AND MOTIVATION

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WHY IS ALL OF THIS IMPORTANT?

Any high-precision measurement/estimation related fields

Has applications in detection of gravitational waves in projects such as LIGO

(LIGO Collaboration, 2011, 2013; Pitkin et al., 2011).

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INTRODUCTION AND MOTIVATION

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SINGLE-SHOT (ns = 1)

1st Shot

Input

MZI

Detection

2nd Shot

MZI

Detection

Input

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INTRODUCTION AND MOTIVATION

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2-SHOTS (ns = 2) (Independent measurements)

1st Shot

2nd Shot

Input

MZI

Detection

Input

MZI

Detection

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INTRODUCTION AND MOTIVATION

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I

MZI

D

CAN EXTEND TO ns-SHOTS (Independent measurements)

I

MZI

D

I

MZI

D

…

ns Shots

Where outcomes at detectors are all considered to be random variables (RV)

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INTRODUCTION AND MOTIVATION

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EXPECTED SCALING

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MATHEMATICAL FRAMEWORK (BAYESIAN ESTIMATION THEORY)

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MATHEMATICAL FRAMEWORK (BAYESIAN ESTIMATION THEORY)

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MATHEMATICAL FRAMEWORK (BAYESIAN ESTIMATION THEORY)

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MATHEMATICAL FRAMEWORK (BAYESIAN ESTIMATION THEORY)

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Before measurement

After measurement

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BAYESIAN ESTIMATION THEORY

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BAYESIAN ESTIMATION THEORY

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(Minimum MSE) MMSE

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BAYESIAN ESTIMATION THEORY

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BAYES THEOREM

Bayes Theorem (for a single shot)

Formula changes for different measurement strategies

Probability of outcome m (evidence)

Prior 𝑝(𝜙)

New information m

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BAYESIAN ESTIMATION THEORY

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BAYES THEOREM

Plugging it in:

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BAYESIAN ESTIMATION THEORY

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BAYES THEOREM

Putting everything together:

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PHYSICAL FRAMEWORK

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PHYSICAL FRAMEWORK (SINGLE SHOT)

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PHYSICAL FRAMEWORK

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INPUT STATE EVOLUTION

lossless

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INPUT STATE EVOLUTION

PHYSICAL FRAMEWORK

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INPUT STATE EVOLUTION

PHYSICAL FRAMEWORK

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INPUT STATE EVOLUTION

PHYSICAL FRAMEWORK

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INPUT STATE EVOLUTION

PHYSICAL FRAMEWORK

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INPUT STATE EVOLUTION

PHYSICAL FRAMEWORK

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INPUT STATE

PHYSICAL FRAMEWORK

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OPTIMAL ONE-SHOT INPUT STATE

N00N

Gaussian

Intermediate

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

No definite analytical expression

PHYSICAL FRAMEWORK

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OPTIMAL ONE-SHOT INPUT STATE

N00N

Gaussian

Intermediate

Optimal in low uncertainty regimes

Optimal in high uncertainty regimes

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

PHYSICAL FRAMEWORK

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NON-ADAPTIVE MEASUREMENT STRATEGY

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EXTENSION TO MULTIPLE SHOTS

Single Shot

Multi-Shot

NON-ADAPTIVE MEASUREMENT STRATEGY

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NON-ADAPTIVE MEASUREMENT STRATEGY

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OPTIMAL TWO-SHOT INPUT STATES

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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NON-ADAPTIVE MEASUREMENT STRATEGY

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GLOBAL OPTIMIZATION (2-SHOT)

Find variance of this entire 2-shot process

Optimize over all the 2(ns)(N+1) variables together

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Minimize:

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NON-ADAPTIVE MEASUREMENT STRATEGY

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GLOBAL OPTIMIZATION (2-SHOT)

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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NON-ADAPTIVE MEASUREMENT STRATEGY

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GLOBAL OPTIMIZATION (2-SHOT)

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Gaussian in high uncertainty regimes

N00N in low uncertainty regimes

True for 3 shots as well, presumably true for ns shots

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NON-ADAPTIVE MEASUREMENT STRATEGY

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LIMITATIONS TO NON-ADAPTIVE STRATEGY: NOT SCALABLE

Becomes unfeasible for larger systems:

2(N+1)(ns) optimization variables

Variance constructed over both shots

1) Bypass optimization. Search for analytical Gaussian and N00N expressions that minimize the variance.

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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NON-ADAPTIVE MEASUREMENT STRATEGY

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ANALYTICAL FORMULA FOR OPTIMAL N00N INPUTS

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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NON-ADAPTIVE MEASUREMENT STRATEGY

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ANALYTICAL FORMULA FOR GAUSSIAN INPUTS

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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NON-ADAPTIVE MEASUREMENT STRATEGY

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ANALYTICAL FORMULA FOR GAUSSIAN INPUTS

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NON-ADAPTIVE MEASUREMENT STRATEGY

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ANALYTICAL FORMULA FOR OPTIMAL GAUSSIAN INPUTS

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NON-ADAPTIVE MEASUREMENT STRATEGY

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BEST-FIT GAUSSIAN AND N00N INPUTS COMPARED TO OPTIMAL

Posterior variance = Prior variance

(No information gain)

RULE: When to use which

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NON-ADAPTIVE MEASUREMENT STRATEGY

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REGIME BOUNDARIES

“Teach” program to distinguish between N00N, Gaussian and Intermediate states
Find boundary using bisection method
Develop a rule of when to use which state

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NON-ADAPTIVE MEASUREMENT STRATEGY

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LIMITATIONS TO NON-ADAPTIVE STRATEGY

Becomes unfeasible for larger systems:

2(N+1)(ns) optimization variables

Variance constructed over both shots

2) Construct variance one shot at a time

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Bypass optimization.

Found analytical expressions and rule

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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LOCAL OPTIMIZATION (2-SHOT)

Find variance of each of the shots individually
Optimize over all the 2(N+1) variables shot-by-shot

Global variance:

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SEMI-ADAPTIVE

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Average variance over the shot outcomes
Posterior uncertainty becomes prior for the 2^nd shot

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SHOT-BY-SHOT OPTIMIZATION

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Semi-adaptive Best-Fit does a great job!

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SEMI-ADAPTIVE REMOVED LIMITATIONS OF NON-ADAPTIVE STRATEGY

Becomes unfeasible for larger systems:

2(N+1)(ns) optimization variables

Variance constructed over both shots

2) Construct variance one shot at a time

Input

MZI

Detection

Input

MZI

Detection

One Shot

Input

MZI

Detection

2nd Shot

Bypass optimization.

Found analytical expressions and rule

SEMI-ADAPTIVE MEASUREMENT STRATEGY

Semi-adaptive is therefore scalable!

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SEMI-ADAPTIVE EXPECTED SCALING

I

M

D

I

M

D

I

M

D

…

ns Shots

Try to predict how much information we gain per shot, on average

Extend to multiple shots

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SEMI-ADAPTIVE EXPECTED SCALING

I

M

D

I

M

D

I

M

D

…

ns Shots

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SEMI-ADAPTIVE EXPECTED SCALING

I

M

D

I

M

D

I

M

D

…

ns Shots

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SEMI-ADAPTIVE MEASUREMENT STRATEGY

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SEMI-ADAPTIVE EXPECTED SCALING

I

M

D

I

M

D

I

M

D

…

ns Shots

Heisenberg scaling

CLT scaling

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ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE MEASUREMENT STRATEGY

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The idea is to make a sequence of adaptive measurements which depend on the outcome of the previous shot.
However, unlike the semi-adaptive sbs we do not average over the possible outcomes.

Again, we can optimize

globally
locally

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EXTENSION TO ADAPTIVE MEASUREMENT

Multi-Shot

Double-Shot, Adaptive

ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE GLOBAL OPTIMIZATION

1^st shot optimal input state

Perform

Measurement

N+1=6 outcomes, with 6 new uncertainties

6 different 2^nd shot optimal input states

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ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE GLOBAL OPTIMIZATION

Optimal inputs for N=5 photons, ns=2 shots for various prior uncertainties

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ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE GLOBAL OPTIMIZATION

Optimal inputs for N=5 photons, ns=2 shots for various prior uncertainties

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ADAPTIVE MEASUREMENT STRATEGY

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ADAPTIVE LOCAL OPTIMIZATION (Feedforward)

Adaptive Best-Fit does a great job!

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ADAPTIVE MEASUREMENT STRATEGY

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COMPARING GLOBAL NON-ADAPTIVE WITH GLOBAL ADAPTIVE

Adaptive case – only marginally better than non-adaptive

To determine which strategy is better

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CONCLUSIONS �

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CONCLUSIONS: OPTIMIZATION VS ANALYTICAL EXPRESSIONS�

Input strategy independent of prev. shot outcome

Input strategy dependent of prev. shot outcome

Global	Most optimal but unscalable

Analytical	Scalable and comparable to global


Local	Scalable but not as good as global non-adaptive
Analytical	Scalable and comparable to global non-adaptive

Global	Most optimal but unscalable
Local	Scalable but not as good as global adaptive
Analytical	Scalable and comparable to global adaptive

Non-adaptive

Semi-adaptive

Adaptive

Need not optimize, use analytical expressions

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CONCLUSIONS: COMPARING DIFFERENT MEASUREMENT STRATEGIES �

Non-adaptive

Semi-adaptive

Adaptive

Better but unscalable. Good for small ns

Not as good but comparable. Scalable, used to develop a scaling model. Good for large ns as well

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CONCLUSIONS �

Non-adaptive

1. Semi-adaptive

Adaptive

More scalable than adaptive

Hard to scale. Marginally better for ns=2. Check ns>2

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CONCLUSIONS �

2. Non-adaptive

1. Semi-adaptive

3. Adaptive

General measurement strategies ranked from best to worst

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REFERENCES

C. M. Caves, Phys. Rev. D 23, 1693 (1981)
Roy S. Bondurant and Jeffrey H. Shapiro Phys. Rev. D 30, 2548;
Bernard Yurke, Samuel L. McCall, and John R. KlauderPhys. Rev. A 33, 4033
Samuel L. Braunstein Phys. Rev. Lett. 69, 3598 (1992);
Jonathan P. Dowling ; Phys. Rev. A 57, 4736 (1998);
M. J. Holland and K. Burnett; Phys. Rev. Lett. 71, 1355 (1993);
B. C. Sanders and G. J. Milburn; Phys. Rev. Lett. 75, 2944 (1995)
A gravitational wave observatory operating beyond the quantum shot-noise limit Nature Physics, 7 (2011);
Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light Nature Photonics, 7 (8) (2013);
Pitkin, M., Reid, S., Rowan, S. et al. Gravitational Wave Detection by Interferometry (Ground and Space). Living Rev. Relativ. 14, 5 (2011)

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THANK YOU FOR YOUR ATTENTION!

- Shreyas (ssadugol@tulane.edu)

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QUESTIONS?

- Shreyas (ssadugol@tulane.edu)

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