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Separability probability formulas and their proofs for generalized two-qubit X-matrices endowed with Hilbert-Schmidt and induced measures. (English) Zbl 1330.15041

Summary: 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 two-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.

MSC:

15B52 Random matrices (algebraic aspects)
81P40 Quantum coherence, entanglement, quantum correlations
33B15 Gamma, beta and polygamma functions

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References:

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