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Physical Review A

Quantum inference methods and hypervirial theorems

Angel Ricardo Plastino, M. Casas, Angel Ricardo Plastino, Antonio Puente

Published Oct 1, 1995• 16 citations
Publisher

Abstract

Arguments of the Jaynes maximum entropy sort have proved to be surprisingly successful in providing one with approximate descriptions of pure states in a variety of scenarios, entirely bypassing any consideration of Schr\"odinger's equation. Thus far, however, the algorithm has been unable to incorporate input information concerning expectation values of complementary observables. We discuss here how to overcome this difficulty by recourse to diagonal hypervirial theorems.

DOI
10.1103/physreva.52.2601

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Arguments of the Jaynes maximum entropy sort have proved to be surprisingly successful in providing one with approximate descriptions of pure states in a variety of scenarios, entirely bypassing any consideration of Schr"odinger's equation. Thus far, however, the algorithm has been unable to incorporate input information concerning expectation values of complementary observables. We discuss here how to overcome this difficulty by recourse to diagonal hypervirial theorems.
The inference of entangled states on the basis of incomplete prior information is usually affected by the problem of fake inferred entanglement. We show that different inferences schemes overestimate the amount of entanglement. However, the Maximum-Entropy Minimum-Norm method yields all the accessible states compatible with the prior information of the system. For bipartite systems the entanglement boundaries are explicitly determined when the knowledge of the system is restricted to the expectation value of a generic operator diagonal in the Bell basis.
We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is shown in a particular example considered recently by Horodecki et al. [Phys. Rev. A 59, 1799 (1999)], by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a ``thermodynamiclike'' description of the entanglement arising out of the maximum entropy principle.
The inference of entangled quantum states by recourse to the maximum entropy (MaxEnt) principle is considered in connection with the recently pointed out problem of fake inferred entanglement (Horodecki R et al 1999 Phys. Rev. A 59 1799). We show that there are operators Â, both diagonal and non-diagonal in the Bell basis, such that, when the expectation value ⟨Â⟩ is taken as prior information, the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of  2 (unlike what happens when the partial information is given by the expectation value of a Bell operator). The fake entanglement generated by the MaxEnt principle is also studied quantitatively by comparing the entanglement of formation of the inferred state with that of the original one.