Likelihood and entropy for quantum tomography

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Likelihood and entropy for quantum tomography

• Z. Hradil, J. Rehácek
• Department of Optics Palacký University,Olomouc
• Czech Republic

• Work was supported by the Czech Ministry of
  Education.

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Collaboration

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

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Outline

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

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Motivation 1
Diffraction on the slit as detection of the
direction
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Measurement according to geometrical
optics propagating rays
Measurement according to the scalar wave theory
diffraction
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• Estimation posterior probability distribution

Fisher information width of post. distribution

• Uncertainty relations

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Motivation 2 Inversion problems
registered mean values j 1, ..M
desired signal i 1, ..N
N number of signal bins (resolution) M number
of scans (measurement)
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Over-determined problems M gt 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 lt N (realm of
physics)
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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

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Motivation 3
All resources are limited!
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Elements of quantum theory
Probability in quantum mechanics
Desired signal density matrix
Measurement positive-valued operator measure
(POVM)
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Complete measurement need not be orthogonal
Generic measurement scans go beyond the space
of the reconstruction
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Stern-Gerlach device
Quantum observables q-numbers
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Mach-Zehnder interferometer
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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.

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

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• Likelihood is the convex functional on the convex
  set of density matrices
• Equation for extremal states

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MaxLik inversion Interpretation
Linear
MaxLik
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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.
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MaxLik Maximum of Relative Entropy
Solution will exhibit plateau of MaxLik states
for under-determined problems (ambiguity)!
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Philosophy 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.
Principle of Maximum Entropy (MaxEnt) selects
the most unbiased solution consistent with the
given constraints. Mathematical formulation
Jaynes
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Entropy Constraints
MaxEnt solution Lagrange multipliers are given
by the solution of the set of nonlinear
constraints
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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.
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MaxEnt assisted MaxLik inversion

• Implementation
• Parametrize MaxEnt solution
• Maximize alternately entropy and likelihood

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MaxEnt assisted MaxLik strategy

• Searching for the worst among the best solutions!

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Interpretation of MaxEnt assisted MaxLik
The plateau of solutions on extended space
Regular part
Classical part
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MaxLik strategy

• Specify the space
• (arbitrary but sufficiently large)
• Find the state
• Specify the space
• Specify the Fisher information matrix F

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Several examples

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

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(Neutron) Transmission tomography

• Exponential attenuation

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Filtered back projection
Maximum likelihood

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

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MaxEnt assisted MaxLik
Numerical simulations using 19 phase scans, 101
pixels each (M1919) Reconstruction on the grid
201x 201 bins (N 40401)
MaxEntLik
Object
MaxLik1
MaxLik2
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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.Rehácek et al.,Multiple-photon resolving
fiber-loop detector, Phys. Rev. A (2003)
061801(R)
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Fiber loop as a multi-channel photon analyser
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Inversion of Bernouli distribution for zero
outcome
Example detection of 2 events 4 channels
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Results of MaxLik inversion
True statistics
(a) Poissonian (b) Composite (d) Gamma (d)
Bose-Einstein
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True statistics 50/50 superposition of
Poissonian statistics with mean numbers1 and
10 Data up to 5 counted events ( 32
channels) Mesh 100
MaxLik
MaxLik MaxLik
Original
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Thank you!