On the "Fake" Inferred Entanglement Associated with the Maximum Entropy Inference of Quantum States

The inference of entangled quantum states by recourse to the maximum entropy principle is considered in connection with the recently pointed out problem of fake inferred entanglement [R. Horodecki, {\it et al.}, Phys. Rev. A {\it 59} (1999) 1799]. We show that there are operators $\hat A$, both diagonal and non diagonal in the Bell basis, such that when the expectation value $<\hat A>$ is taken as prior information the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of $\hat A^2$ (unlike what happens when the partial information is given by the expectation value of a Bell operator). The fake entanglement generated by the maximum entropy principle is also studied quantitatively by comparing the entanglement of formation of the inferred state with that of the original one.