Determinantal probability densities at two-qubit separability-entanglement boundary and associated Fisher information
The determinants (|rho^PT|) of the partial transposes of 4 x 4 density matrices (rho) have possible values in the interval [-1/16, 1/256], and are nonnegative if and only if rho is separable. In arXiv:1301:6617, we reported a "concise" formula for the cumulative/separability Hilbert-Schmidt probability P(alpha) of |rho^PT| over the nonnegative subinterval [0, 1/256] where alpha is a Dyson-index-like parameter, with alpha = 1/2 denoting the 9-dimensional generic two-rebit systems, alpha = 1, the 15-dimensional generic two-qubit systems,.... Here, we seek to expand our understanding of the underlying probability distributions p_alpha(|rho^PT|) over [-1/16, 1/256] by determining their y-intercepts--that is the values at |rho^PT| = 0 (the separability-entanglement boundary). Our numerical evidence strongly indicates that p_2(0) = 7425 / 34, p_3(0)= 7696 / 69, and possibly, that p_1(0) = 390. The first derivative p_alpha^'(0) is positive for alpha = 1/2 and 1, but negative for alpha > 1. We also estimate the Fisher information (declining with alpha) of the alpha-parameterized family of probability distributions p_alpha(|rho^PT|).