Quantum estimation via the minimum Kullback entropy principle

We address quantum estimation in situations where one has at disposal data from the measurement of an incomplete set of observables and some a priori information on the state itself. By expressing the a priori information in terms of a bias toward a given state, the problem may be faced by minimizing the quantum relative entropy (Kullback entropy) with the constraint of reproducing the data. We exploit the resulting minimum Kullback entropy principle for the estimation of a quantum state from the measurement of a single observable, either from the sole mean value or from the complete probability distribution, and apply it as a tool for the estimation of weak Hamiltonian processes. Qubit and harmonic oscillator systems are analyzed in some detail.

[1]  Approximate angular momentum projection using the maximum entropy principle , 1992 .

[2]  Dmitri Mogilevtsev,et al.  Diagonal element inference by direct detection , 1998 .

[3]  Z. Hradil.  Quantum-state estimation , 1996, quant-ph/9609012.

[4]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[5]  M. Donald.  On the relative entropy , 1986 .

[6]  Konrad Banaszek,et al.  Photon counting with a loop detector. , 2003, Optics letters.

[7]  F. Komaki,et al.  Bayesian predictive density operators for exchangeable quantum-statistical models , 2005 .

[8]  G. Lindblad.  Entropy, information and quantum measurements , 1973 .

[9]  Matteo G. A. Paris,et al.  A two-step MaxLik-MaxEnt strategy to infer photon distribution from on/off measurement at low quantum efficiency , 2005 .

[10]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[11]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[12]  G. Bilbro,et al.  Mean-field approximation minimizes relative entropy , 1991 .

[13]  O. Haderka,et al.  Experimental multi-photon-resolving detector using a single avalanche photodiode , 2003, quant-ph/0302154.

[14]  Heinz Mühlenbein,et al.  The Estimation of Distributions and the Minimum Relative Entropy Principle , 2005, Evol. Comput..

[15]  S. Braunstein.  GEOMETRY OF QUANTUM INFERENCE , 1996 .

[16]  T. Ulrych,et al.  Minimum relative entropy: Forward probabilistic modeling , 1993 .

[17]  Marco Genovese,et al.  Experimental reconstruction of photon statistics without photon counting. , 2005, Physical review letters.

[18]  Tryphon T. Georgiou.  Relative entropy and the multivariable multidimensional moment problem , 2006, IEEE Transactions on Information Theory.

[19]  T. Ulrych,et al.  Minimum Relative Entropy Inversion: Theory and Application to Recovering the Release History of a Groundwater Contaminant , 1996 .

[20]  V. Vedral.  The role of relative entropy in quantum information theory , 2001, quant-ph/0102094.

[21]  P. W. Lamberti,et al.  Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states , 2005, quant-ph/0508138.

[22]  A. R. Plastino,et al.  MINIMUM KULLBACK ENTROPY APPROACH TO THE FOKKER-PLANCK EQUATION , 1997 .

[23]  H. Umegaki.  Conditional expectation in an operator algebra. IV. Entropy and information , 1962 .

[24]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[25]  M. G. A. Paris,et al.  Measuring the photon distribution with ON/OFF photodetectors , 2006 .

[26]  C. Caves,et al.  Quantum Bayes rule , 2000, quant-ph/0008113.

[27]  P. Slater.  COMPARATIVE NONINFORMATIVITIES OF QUANTUM PRIORS BASED ON MONOTONE METRICS , 1997, quant-ph/9703012.

[28]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[29]  Stefano Olivares,et al.  Photon statistics without counting photons , 2004, quant-ph/0405139.

[30]  Matteo G. A. Paris,et al.  Quorum of observables for universal quantum estimation , 2001 .

[31]  Vladimír Bužek,et al.  Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography , 1998 .

[32]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[33]  J. Rehacek,et al.  Multiple-photon resolving fiber-loop detector , 2003 .

[34]  Zdenek Hradil,et al.  Quantum theory of incompatible observations , 1999, quant-ph/9911067.

[35]  M. Donald,et al.  A priori probability and localized observers , 1992 .