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      Formulas for Generalized Two-Qubit Separability Probabilities

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          Abstract

          To begin, we find certain formulas \(Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)\), for \(k = -1, 0, 1,...,9\). These yield that part of the total separability probability, \(P(k,\alpha)\), for generalized (real, complex, quaternionic,\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality \(|\rho^{PT}| >|\rho|\) holds. Here \(\rho\) denotes a \(4 \times 4\) density matrix, obtained by tracing over the pure states in \(4 \times (4 +k)\)-dimensions, and \(\rho^{PT}\), its partial transpose. Further, \(\alpha\) is a Dyson-index-like parameter with \(\alpha = 1\) for the standard (15-dimensional) convex set of (complex) two-qubit states. For \(k=0\), we obtain the previously reported Hilbert-Schmidt formulas, with (the real case) \(Q(0,\frac{1}{2}) = \frac{29}{128}\), (the standard complex case) \(Q(0,1)=\frac{4}{33}\), and (the quaternionic case) \(Q(0,2)= \frac{13}{323}\)---the three simply equalling \( P(0,\alpha)/2\). The factors \(G_2^k(\alpha)\) are sums of polynomial-weighted generalized hypergeometric functions \(_{p}F_{p-1}\), \(p \geq 7\), all with argument \(z=\frac{27}{64} =(\frac{3}{4})^3\). We find number-theoretic-based formulas for the upper (\(u_{ik}\)) and lower (\(b_{ik}\)) parameter sets of these functions and, then, equivalently express \(G_2^k(\alpha)\) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for \(P(0,\alpha) =2 Q(0,\alpha)\). For nonnegative half-integer and integer values of \(\alpha\), \(Q(k,\alpha)\) has descending roots starting at \(k=-\alpha-1\). Then, we (C. Dunkl and I) construct a remarkably compact (hypergeometric) form for \(Q(k,\alpha)\) itself. The possibility of an analogous "master" formula for \(P(k,\alpha)\) is, then, investigated, and a number of interesting results found.

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          Author and article information

          Journal
          2016-09-27
          Article
          1609.08561
          c2368631-05d7-408a-8c64-a5f2ecb18a64

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          81P45, 33C20, 65Q10, 62E17, 62E20
          67 pages, 6 figures, 9 appendices
          quant-ph math-ph math.MP math.PR

          Mathematical physics,Quantum physics & Field theory,Mathematical & Computational physics,Probability

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