Two-Qubit Rational-Valued Entanglement-Boundary Probability Densities and a Fisher Information Equality Conjecture

Abstract

We consider a pair of one-parameter (alpha) families of generalized two-qubit determinantal Hilbert-Schmidt probability distributions, p_{alpha}(|rho^{PT}|) and q_{alpha}(|rho|), where rho is a 4 x 4 density matrix, rho^{PT}, its partial transpose, with |rho^{PT}| \in [-1/16,1/256] and |rho| \in [0, 1/256]. The Dyson-index-like (random matrix) parameter alpha is 1/2 for the 9-dimensional generic two-rebit systems, 1 for the 15-dimensional generic two-qubit systems,... Numerical (moment-based probability-distribution-reconstruction) analyses suggest the conjecture that the Fisher information--a measure over alpha--is identical for the two distinct families. Further, we study the values of p_{alpha}(0), the probability densities at the separability-entanglement boundary, with evidence strongly indicating that p_2(0) =7425/34 and p_3(0)= 7696/69. Despite extensive results of such a nature, we have not yet succeeded--in contrast with the corresponding rational-valued separability probabilities (arXiv.org:1301.6617)--in generating an underlying, explanatory ("concise") formula for p_{\alpha}(0), even though the corresponding denominators of both sets of rational-valued results are closely related, having almost identical (small) prime factors. The first derivatives p_{alpha}^{'}(0) are positive for alpha = 1/2 and 1, but negative for alpha > 1.