Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures

TL;DR

This paper derives exact formulas for the probability that generalized two-qubit X-matrices are separable under Hilbert-Schmidt and induced measures, providing rigorous analytical results and geometric insights.

Contribution

It provides the first exhaustive collection of separability probability formulas for generalized two-qubit X-matrices with rigorous proofs and geometric interpretations.

Findings

Explicit formulas for separability probabilities under various measures

Analytical parallels to earlier heuristic results for general 2-qubit states

Geometric interpretations of the Dyson-index-like parameter

Abstract

Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert-Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general 2-qubit states--deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter $\alpha$ employed.