Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Abstract

Previously, a formula, incorporating a $5F4$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $⟨{\left|{\rho }^{PT}\right|}^n{\left|\rho \right|}^k⟩/⟨{\left|\rho \right|}^k⟩$ of $4\times 4$ density-matrices ($\rho$), and their partial transposes (${\rho }^{PT}|$) was applied with $k=0$ to the generalized two-qubit separability-probability question. The formula can, further, be viewed we note here, as an averaging over "induced measures in the space of mixed quantum states". The associated induced-measure separability probabilities ($k=1,2,\dots$) are found--{\it via} a high-precision density approximation procedure--to assume interesting, relatively simple rational values in the two-re[al]bit ($\alpha =\frac{1}{2}$), (standard) two-qubit ($\alpha =1$) and two-quater[nionic]bit ($\alpha =2$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,\ldots) formulas that successfully reproduce the rational values assumed for {\it general} $k$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

18 pages, 2 figures, revised version to appear in Advances in Mathematical Physics

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• 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
• Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems
• Confident entanglement detection via separable numerical range