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AIP conference proceedings

Inference of quantum states: Maximum entropy and “fake” inferred entanglement

Josep Amengual i Batle

Published Jan 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1477055

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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.
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.
We present an approximate inference approach based on the Information Theory and the Pseudoinverse Formalism that allows to obtain all the accessible micro-states of a quantum system compatible with a given information. We assume that the Hamiltonian of the system is unknown and the only available information is restricted to a set of expectation values of noncommuting operators. With this information the density matrix is inferred and, exploring the Null space of the input operators, one can define the physical regions for which the density matrix is positive definite. The method is applied for the inference of entangled quantum states.
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.
We examine the inference of quantum density operators from incomplete information by means of the maximization of general nonadditive entropic forms. Extended thermodynamic relations are given. When applied to a bipartite spin 1 / 2 system, the formalism allows one to avoid fake entanglement for data based on the Bell-Clauser-Horne-Shimony-Holt observable, and, in general, on any set of Bell constraints. Particular results obtained with the Tsallis entropy and with an introduced exponential entropic form are also discussed.