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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
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
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
• 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
)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
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
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)
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