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arXiv:quant-ph/0107023 · Journal of Physics A Mathematical and General

On the `fake' inferred entanglement associated with the maximum entropy inference of quantum states

J Batle, M. Casas, A R Plastino, A Plastino

Published Aug 10, 2001• 2 citations
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Abstract

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.

DOI
10.1088/0305-4470/34/33/309

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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.
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  is taken as prior information, the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of Â2 (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.
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.
We study a particular example considered by Horodecki et al (1999 Phys. Rev. A 59 1799), concerning the statistical inference of quantum entanglement using the Jaynes principle. Assume a Clauser–Horne–Simony–Holt (CHSH) Bell operator, a sum of two operators . Given only an average of the Bell–CHSH operator, we may overestimate entanglement. However, the estimated entanglement is decreased (never increases) when we use the expectation value of the operator X as additional information. A minimum entanglement state is obtained by minimizing the variance of the observable X .
We show, by explicit examples, that the Jaynes principle when applied to compound quantum systems may produce the entangled maximum entropy states compatible with data coming from nonentangled (separable) states. It means that the Jaynes statistical inference scheme may lead to a wrong conclusion about entanglement, which is a crucial parameter in quantum information theory. We suggest that in all the processes where entanglement is needed, the proper inference scheme should involve minimization of entanglement. Examples illustrating the proposed scheme are provided.