Tomography for quantum diagnostics

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Tomography for quantum diagnostics

• Z. Hradil, J. Rehácek
• Department of Optics, Palacký University
• Olomouc, Czech Rep.
• D. Mogilevtsev
• Institute of Physics, Belarus National Academy of
  Sciences, Minsk, Belarus

Supported by the Czech Ministry of Education and
EU project FP 6 COVAQIAL
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Outline

• Motivation
• Quantum measurement and estimation
• Objective tomography for diagnostics
• Examples Homodyne tomography
• Summary

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MotivationInverse problems
Ij Sk cjk mk ..detected mean values
j 1,2,M mk signal
k 1,2,N

Over-determined problems M gt N Well defined
problems M N Under-determined
problems M lt N
Quantum tomography with continuous
variables always under-determined problem
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Motivation Fidelity ?!?

• 
  
• Is the fidelity 99 good or bad?
• f lt0.91gt2 exp(-0.01) ? 1-0.01
• Coherent states 0.9gt (estimate) and 1gt (true)
  give this fidelity, though the difference in
  energies is about 20 !?!?

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Our goal To establish quantum tomography as an
objective tool for diagnostics of quantum
systems

• Objective tomography scheme
• Reconstruction is not equally good in the full
  Hilbert space Field of view defines the visible
  part of the Hilbert space
• How to reconstruct and where to reconstruct are
  NOT independent tasks in generic tomographic
  schemes
• Errors matter!

Hradil, Mogilevtsev, Rehacek, Biased tomography
schemes an objective approach, PRL 96, 230401
(2006).
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Elements of quantum theory
Probability in QM pj Tr(? Aj) Measurement
elements of positive-valued operator measure
(POVM) Aj 0 Signal density matrix ?
0 Generic over-complete measurement ?j Aj G
0 may always be cast in the form of POVM ?j
G-1/2 Aj G-1/2 1G
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Geometry Overlap of states
Projector Ai yigtltyi
Overlap of all projectors Si ltyijgt2
Maximum overlap ?Si yigtltyi?jgt ljgt
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Tomography based on quantum estimation

• Reconstructed state ? is treated as a set of
  parameters
• Set of tomographically complete measurements
  needed

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Maximum Likelihood reconstruction

• Maximum Likelihood (MaxLik) principle is not a
  rule that requires justification Bet Always On
  the Highest Chance!
• Likelihood L quantifies the degree of belief in
  certain hypothesis under the condition of the
  given data.
• MaxLik principle selects the most likely
  configuration
• Information is updated according to Bayes rule
• prior probability posterior
  probability
• P(?D) P(D?) p(?) p(D)-1

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Generic reconstruction scheme
Log-likelihod for generic measurement log L ?i
Nj log pj / (?k pk) (probabilities are mutually
normalized) Equivalent formulation estimation
of parameters with Poissonian probabilities and
unknown mean l (constrained MaxLik by
Fermi) log L ?j Nj log (l pj ) - l ?j pj
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Extremal equation
R r G r R (Sjpi) /(SjNi) S (Nk/pk) Ak G Si
Ai
RG rG rG RG G-1/2RG-1/2, rG G1/2 r
G1/2 Solution in the iterative form rGRGrGRG
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Likelihood is convex functional defined on the
convex manifold of density matrices
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MaxLik in terms of Quantum Mechanics

• Fluctuations in k-th channel
• (Dek)2 Tr(rAk) 1- Tr(rAk)
• All observations cannot be equally trusted!
• MaxLik estimation in 3 steps
• Re-define POVM elements Ak ? mk Ak
• Postulate mean values mkTr(rAk) fk
• Postulate closure relation (G1)
• Sk mkAk 1

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MaxLik interpretation

• Linear inversion
• Sk Ak ? 1 Tr(rAk) fk

MaxLik inversion Sk Ak 1G
Tr(r Ak) ? fk where Ak (fk/pk) Ak
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Field of view in tomography
Since r G-1/2 rG G-1/2 the solution must be
searched in the subspace spanned by nonzero
eigenvalues of G only! No data binning is needed!
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Analogies in optics People with better optics
should see better!
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Optical transfer function
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Error analysis

• MaxLik reconstruction characterizes the estimated
  state as random variable
• Any prediction based on tomography is uncertain
  (e.g. fidelity, Wigner function at origin, etc.)
• Q ltQgtML DQ

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• Reconstruction result is not just a single state
  but the family of the states described by
  posterior distribution (Bayes rule)
• Performance measure linear in quantum state
  (fidelity, Wigner function at origin) z
  Tr(Zr)
• z Tr(ZrML) ltzF-1zgt1/2
• F Fisher information matrix
• zgt vector with components corresponding to
  Z in the fixed operator basis
• Example Z Sn (-1)n ngtltn gives the Wigner
  function
• at origin

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Uncertainty for tomography
Tomography never tells everything due to the
uncertainty relation
ltzF-1zgt ltwF-1wgt ? ltwF-1zgt2
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Strategy of objective tomography

• Analysis of measurement field of view is
  defined by eigenbasis of G
• Iterative solution of MaxLik equations
• Characterization of errors

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Several examples

• Phase estimation
• Transmission tomography
• Reconstruction of photocount statistics
• Image reconstruction
• Vortex beam analysis
• Quantification of entanglement
• Reconstruction of neutron wave packet
• Reconstruction based on homodyne detection
• Full reconstruction based on on/off detection

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Reconstruction of Wigner function
Homodyne measurement
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Recent results-simulations of realistic homodyne
tomography

• Standard setup used for detection of negative
  Wigner function
• 6 phase cuts in phase space, efficiency ?0,8
• 1,2 .105 detected events accumulated into 64 bins
  for each phase cut
• ML estimation using 1000 iterations
• Simulation repeated 1000 times

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Homodyne tomography for several phase cuts
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Traditional interpretation-reconstruction of
cat-states
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Tomography revisited-effect of dimension
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Tomography revisited-detected events
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Tomography revisited-efficiency
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? Good news Reconstruction can be tuned to the
desired resolution, objective tomography gives
the tools!
? Bad news Homodyne tomography is not able to
reconstruct the unknown state!
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Rationale behind If the field of view is too
large the states r andre lime?0 (1-e) r
ea/?egtlta/? ecannot be distinguished on the
basis of homodyne tomography.
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Why errors cannot be always simulatedAssume
the statistics of variable s estimated on the
basis of N trials s (1/N) Si
xi2Singular Statistics p(x) 1/p
sinc2x ltsgttheory ? but ltsgtexp ? N1/2 ,
lts2gtexp ? N3/2 and SNR ? N1/4
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Summary

• Any piece of measured information may be
  quantified by means of objective tomography
  scheme
• Biased scheme various parts of quantum state
  are observed (reconstructed) with different
  accuracy
• Operator G plays the role of optical transfer
  function for quantum tomography
• Errors are important !

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• For more details see the poster
• J.Rehácek, D. Mogilevtsev, Z.Hradil
• Is objective quantum homodyne tomography
  possible?

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The End