Additional information decreases the estimated entanglement using the Jaynes principle

TL;DR

This paper investigates how incorporating additional information about a quantum observable affects the estimated entanglement using the Jaynes principle, showing that more information decreases the estimated entanglement.

Contribution

It demonstrates that adding expectation values of observables reduces the estimated entanglement, refining the inference method based on the Jaynes principle in quantum systems.

Findings

Additional information decreases estimated entanglement

Using variance minimization yields minimal entanglement states

Inclusion of observable expectations refines entanglement estimates

Abstract

We study a particular example considered in {[Phys. Rev. A {\bf 59,} 1799 (1999)]}, 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 $\sqrt{2}(X+Z)$. 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 $X$.