Additional information decreases the estimated entanglement using the Jaynes 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.

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