Likelihood and entropy for quantum tomography Z. Hradil, J. Řeháček Department of Optics Palacký University,Olomouc Czech Republic Work was supported by.

Likelihood and entropy for quantum tomography

Z. Hradil, J. Řeháček Department of Optics Palacký

University,OlomoucCzech Republic

Work was supported by the Czech Ministry of Education.

Collaboration

•SLO UP ( O. Haderka) •Vienna: A. Zeilinger, H. Rauch, M. Zawisky•Bari: S. Pascazio•Others: HMI Berlin, ILL Grenoble

•Motivation •Inverse problems •Quantum measurements vs. estimations •MaxLik principle •MaxEnt principle •Several examples•Summary

Outline

Diffraction on the slit as detection of the direction

Motivation 1:

2( ) ( 1 ) 2 , ( 2 )sincP a ak dpx

Measurement according to geometrical optics:propagating rays

sinhxpd

Measurement according to the scalar wave theory: diffraction

Estimation: posterior probability distribution

( ) Gaussian approximationpost pP

Fisher information: width of post. distribution

2

logd

p pF d

Uncertainty relations

x p = 2 , p = 1 F

,x p pp p pp

j jiicI

registered mean values

j = 1, ..M

desired signal

i= 1, ..N

N number of signal bins (resolution)M number of scans (measurement)

Motivation 2: Inversion problems

Over-determined problems M > N(engineering solution: credible interpretation)

Well defined problems M = N (linear inversion may appear as

ill posed problem due to the imposed constraints)

Under-determined problems M < N(realm of physics)

Inversion problems: Tomography

Medicine: CT, NMR, PET, etc.:

nondestructive visualization of 3D objects

Back-Projection (Inverse Radon transform)

●ill-posed problem

●fails in some applications

Motivation 3:

All resources are limited!

Elements of quantum theory

0

T(r )p ii

Probability in quantum mechanics

Desired signal: density matrix

Measurement: positive-valued operator measure (POVM)

0i

1 ii

Complete measurement: need not be orthogonal

Generic measurement: scans go beyond the space of the reconstruction

1/ 2 1/ 20 1

i i Gii

G G G

Quantum observables: q-numbers

Stern-Gerlach device

Mach-Zehnder interferometer

• Maximum Likelihood (MaxLik) principle selects the most likely configuration

• Likelihood L quantifies the degree of belief in certain hypothesis under the condition of the given data.

log log (ρ)pf i i i L

Principle of MaxLik

MaxLik principle is not a rule that requires justification. Mathematical formulation: Fisher

Bet Always On the Highest Chance!

Philosophy behind

MaxLik estimation

• Measurement: prior info posterior info• Bayes rule:

• The most likely configuration is taken as the result of estimation

• Prior information and existing constraints can be easily incorporated

p D|ρp ρ|D p ρ

p D

• Likelihood is the convex functional on the convex set of density matrices

• Equation for extremal states

i i( )pfR ji

R RR1R or or

1i i f i)(Tr i

1i i f i)(Tr i ii

ii )(p

f

MaxLik

Linear

MaxLik inversion: Interpretation

)p1(Np)n(2 Various projections are

counted with different accuracy.

Accuracy depends on the unknown quantum state.

Optimal estimation strategy must re-interpret the registered data and estimate the state simultaneously.

Optimal estimation should be nonlinear.

f logf)(p logf )|(log ii iii i pfL

MaxLik = Maximum of Relative Entropy

Solution will exhibit plateau of MaxLik states for under-determined problems (ambiguity)!

Laplace's Principle of Insufficient Reasoning: If there is no reason to prefer among several possibilities, than the best strategy is to consider them as equally likely and pick up the average.

Principle of Maximum Entropy (MaxEnt) selects the most unbiased solution consistent with the given constraints.Mathematical formulation: Jaynes

Philosophy behind Maximum Entropy

MaxEnt solution

Lagrange multipliers are given by the solution ofthe set of nonlinear constraints

-1iiii iiρ = exp( )[Tr exp( ) ] AA

i iTr[ρ( ) ]f A

S = - Tr( ρ log ρ )Entropy

Constraints i( )iTr ρf A

MaxLik: the most optimistic guess.

Problem: Ambiguity of solutions!

MaxEnt: the most pesimistic guess.

Problem: Inconsistent constraints.

Proposal: Maximize the entropy over the convex set of MaxLik states! Convexity of entropy will guarantee the uniqueness of the solution. MaxLik will make the all the constraints consistent.

Implementation

• Parametrize MaxEnt solution

• Maximize alternately entropy and likelihood

MaxEnt assisted MaxLik inversion

MaxEnt assisted MaxLik strategy

Searching for the worst among the best solutions!

Interpretation of MaxEnt assisted MaxLik

The plateau of solutions on extended space

1est

Regular part “Classical” part

MaxLik strategy

•Specify the space (arbitrary but sufficiently large) •Find the state

•Specify the space

•Specify the Fisher information matrix F

MaxLikH

MaxLik

H

•Phase estimation•Reconstruction of Wigner function•Transmission tomography•Reconstruction of photocount statistics•Image reconstruction •Vortex beam analysis•Quantification of entanglement•Operational information

Several examples

(Neutron) Transmission tomography

• Exponential attenuation

I h , I 0 eh ,

x , y d

Filtered back projection

Maximum likelihood

J. Řeháček, Z. Hradil, M. Zawisky, W. Treimer, M. Strobl: Maximum Likelihood absorption tomography,  Europhys. Lett. 59 694- 700 (2002).

MaxEnt assisted MaxLik

Numerical simulations using 19 phase scans, 101 pixels each (M=1919)

Reconstruction on the grid 201x 201 bins (N= 40401)

ObjectMaxLik1 MaxLik2

MaxEnt+Lik

Fiber-loop detector

• Commercially available single-photon detectors do not have single-photon resolution

• Cheap (partial) solution: beam splitting

• Coincidences tell us about multi-photon content

J.Řeháček et al.,Multiple-photon resolving fiber-loop detector, Phys. Rev. A (2003) 061801(R)

Fiber loop as a multi-channel photon analyser

0m mm0

)1( mp

Example: detection of 2 events = 4 channels

00 1 21 T ( 1 T)pm

m m

Inversion of Bernouli distribution for zero outcome

10 0021 ( 1 T)p pm

m m

01 0011 T)p pm

m m

11 00 10 011p p p p

Results of MaxLik inversion:

True statistics:

(a) Poissonian

(b) Composite

(d) Gamma

(d) Bose-Einstein

True statistics: 50/50 superposition of Poissonian statistics with mean numbers1 and 10Data: up to 5 counted events (= 32 channels)Mesh: 100

Original MaxLik MaxLik & MaxLik

Thank you!