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arXiv:quant-ph/9701038 · Journal of Modern Optics

Reconstruction of quantum states of spin systems via the Jaynes principle of maximum entropy

Vladimír Bužek, G. Drobný, G. Adam, R. Derka, P. L. Knight

Published Nov 1, 1997• 28 citations
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Abstract

We apply the Jaynes principle of maximum entropy for the partial reconstruction of correlated spin states. We determine the minimum set of observables which is necessary for the complete reconstruction of the most correlated states of systems composed of spins 1/2 (e.g. the Bell and the Greenberger-Horne-Zeilinger states). We investigate to what extent an incomplete measurement can reveal non-classical features of correlated spin states.

DOI
10.1080/09500349708231905

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We apply the Jaynes principle of maximum entropy for the partial reconstruction of correlated spin states. We determine the minimum set of observables which is necessary for the complete reconstruction of the most correlated states of systems composed of spins 1/2 (e.g. the Bell and the Greenberger-Horne-Zeilinger states). We investigate to what extent an incomplete measurement can reveal non-classical features of correlated spin states.
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A concern has been expressed that ``the Jaynes principle can produce fake entanglement'' [R. Horodecki et al., Phys. Rev. A 59, 1799 (1999)]. In this paper we discuss the general problem of distilling maximally entangled states from N copies of a bipartite quantum system about which only partial information is known, for instance, in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principle of maximum entropy to more than one copy of a system, but the nature of this problem is classical and was discussed extensively by Jaynes. Under the additional assumption that the state ${\ensuremath{\rho}}^{(N)}$ of the N copies of the quantum system is exchangeable, one can write down a simple general expression for ${\ensuremath{\rho}}^{(N)}.$ By measuring one or more of the subsystems, one can gain information and update the state estimate for the remaining subsystems with the quantum version of the Bayes rule. Using this rule, we show how to modify two standard entanglement purification protocols, one-way hashing and recurrence, so that they can be applied to exchangeable states. We thus give an explicit algorithm for distilling entanglement from an unknown or partially known quantum state.
We apply the Jaynes principle of maximum entropy [Phys. Rev. 106, 620 (1957); 108, 171 (1957)] for a reconstruction of Wigner functions of quantum-mechanical states of light on different observation levels. We study how quantum interference between components of superpositions of coherent states, which is responsible for the appearance of nonclassical effects, can be detected on different observation levels. We analyze in detail the reconstruction of Wigner functions of squeezed states on different observation levels in the case of nonunit detection efficiency modeled as a decay of the state under consideration into a zero-temperature reservoir. \textcopyright{} 1996 The American Physical Society.
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.