Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors

Abstract

The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (σ_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_α). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},σ_{1}^2,α) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < α< \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featured. Our model can be contrasted with the three-parameter (b_{q}, σ_{q}^2,q) one of Abe and Rajagopal, which employs a q(Tsallis)-parameter rather than $\alpha$, and has simply q-invariant HS separability probabilities of 1/2. Our results emerge in a study initially focused on embedding certain information metrics over the two-level quantum systems into a q-framework. We find evidence that Srednicki's recently-stated biasedness criterion for noninformative priors yields rankings of priors fully consistent with an information-theoretic test of Clarke, previously applied to quantum systems by Slater.

26 pages, 12 figures

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