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Formulas for rational-valued separability probabilities of random induced generalized two-qubit states. (English) Zbl 1337.81027
Summary: Previously, a formula, incorporating a $$5 F 4$$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $$\langle\left|\rho^{\text{P} \text{T}}\right|^n \left|\rho\right|^k\rangle /\langle\left|\rho\right|^k\rangle$$ of $$4 \times 4$$ density-matrices ($$\rho$$) and their partial transposes ($$| \rho^{\mathrm{PT}}|$$), was applied with $$k = 0$$ to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states”. The associated induced-measure separability probabilities ($$k = 1,2, \ldots$$) are found – via a high-precision density approximation procedure – to assume interesting, relatively simple rational values in the two-re[al]bit ($$\alpha = 1 / 2$$), (standard) two-qubit ($$\alpha = 1$$), and two-quater[nionic]bit ($$\alpha = 2$$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,…) formulas that successfully reproduce the rational values assumed for general $$k$$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

##### MSC:
 81P40 Quantum coherence, entanglement, quantum correlations 33C05 Classical hypergeometric functions, $${}_2F_1$$
DLMF
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