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

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.

20 pages, two formulas reformatted for presentational purposes

References in corpus (4)

• Entanglement universality of two-qubit X-states
• Volumes of conditioned bipartite state spaces
• Numerical Computations of Separability Probabilities
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities

Cited by in corpus (5)

• Maximally genuine multipartite entangled mixed X-states of N-qubits
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures