Z. Hradil, J. Řeháček

Transcription

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

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

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

4 Motivation 1: Diffraction on the slit as detection of the direction

5 Measurement according to geometrical optics: propagating rays h d px sin Measurement according to the scalar wave theory: diffraction P( ) ( 1 ) sinc2 a p x 2h, ( ak 2d )

6 Estimation: posterior probability distribution P post p( ) Gaussian approximation Fisher information: width of post. distribution F p d log p d 2 Uncertainty relations x p, p pp p p x p = h 2, p =1 F

7 Motivation 2: Inversion problems I j cji i registered mean values j = 1,..M desired signal i= 1,..N N number of signal bins (resolution) M number of scans (measurement)

8 Over-determined problems M>N (engineering solution: credible interpretation) Well defined problems M=N (linear inversion may appear as l posed problem due to the imposed constrain Under-determined problems (realm of physics) M<N

9 Inversion problems: Tomography Medicine: CT, NMR, PET, etc.: nondestructive visualization of 3D objects Back-Projection (Inverse Radon transform) ill-posed fails problem in some applications

10 Motivation 3: All resources are limited!

11 Elements of quantum theory Probability in quantum mechanics pi T(r i ) Desired signal: density matrix 0 Measurement: positive-valued operator measure (POVM) i 0

12 Complete measurement: need not be orthogona i i 1 Generic measurement: scans go beyond the space of the reconstruction G i i 0 G G 1/2 i 1/2 i 1G

13 Stern-Gerlach device Quantum observables: q-numbers

14 Mach-Zehnder interferometer

15 Principle of MaxLik 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. logl f i logpi(ρ) i

16 Philosophy behind Bet Always On the Highest Chance! MaxLik principle is not a rule that requires justification. Mathematical formulation: Fisher

17 MaxLik estimation Measurement: prior info posterior info p D ρ p ρ D p ρ Bayes rule: p D The most likely configuration is taken as the result of estimation Prior information and existing constraints can be easily incorporated

18 Likelihood is the convex functional on the convex set of density matrices Equation for extremal states Rρ=ρ R or R i fi = 1ρ or p(i ) j RρR=ρ

19 MaxLik inversion: Interpretation Linear i Π i 1 Tr(ρ Π i) = f i MaxLik i π i = 1ρ Tr(ρ π i) f i πi = fi Π i pi (ρ )

20 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 2 ( n) = Np(1 p)

21

22

23 MaxLik = Maximum of Relative Entropy logl (f p) = i f i logpi (ρ ) i f i logf i Solution will exhibit plateau of MaxLik states for under-determined problems (ambiguity)!

24 hilosophy behind Maximum Entropy 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. rinciple of Maximum Entropy (MaxEnt) selects he most unbiased solution consistent with he given constraints. Mathematical formulation: Jaynes

25 S = - Tr(ρ log ρ ) ( A i) Constraints f Trρ Entropy i MaxEnt solution -1 ρ =[Tr exp( i ia i ) ] exp( i ia i ) Lagrange multipliers are given by the solution the set of nonlinear constraints f Tr[ρ( )A i ] i

26 MaxLik: the most optimistic guess. MaxEnt: the most pesimistic guess. Problem: Inconsistent Problem: Ambiguity of constraints. solutions! 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

27 MaxEnt assisted MaxLik inversion Implementation Parametrize MaxEnt solution Maximize alternately entropy and likelihood

28 MaxEnt assisted MaxLik strategy earching for the worst among the best solution

29 Interpretation of MaxEnt assisted MaxLik The plateau of solutions on extended space est 1 Regular part Classical part

30 MaxLik strategy Specify the space H (arbitrary but sufficiently large) Find the state Specify the space H MaxLik MaxLik Specify the Fisher information matrix F

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

32 (Neutron) Transmission tomography Exponential attenuation I h, I 0 e x,y d h,

33 Filtered back projection Maximum likelihood J. Řeháček, Z. Hradil, M. Zawisky, W. Treimer, M. Strobl: Maximum Likelihood absorption tomography, Europhys. Lett (2002).

34 MaxEnt assisted MaxLik merical simulations using 19 phase scans, 101 pixels each (M=19 onstruction on the grid 201x 201 bins (N= 40401) Object MaxLik1 MaxLik2 MaxEnt+Lik

35 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 Řeháček et al.,multiple-photon resolving fiber-loop detector, Phys. Rev. A (2003) (R

36 Fiber loop as a multi-channel photon analyser

37 nversion of Bernouli distribution for zero outco m ( 1 η ) m=0 ρ mm p0 = Example: detection of 2 events = 4 p 00 p 10 p 01 p m 1 1T (2 m 1 (2 m m 1 1T) m m 11 1 T) m p00 1 p00 p10 p01 1 T) m p00 m channels

38 Results of MaxLik inversion: True statistics: (a) Poissonian (b) Composite (d) Gamma (d) BoseEinstein

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

40 Thank you!