Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

Abstract

We begin by investigating relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit "separability functions"--those recently advanced by Lovas and Andai (J. Phys. A 50 [2017] 295303), and those earlier presented by Slater (J. Phys. A 40 [2007] 14279). In the Lovas-Andai framework, the independent variable $\epsilon \in [0,1]$ is the ratio $\sigma (V)$ of the singular values of the $2\times 2$ matrix $V=D_2^{1/2}{D}_1^{-1/2}$ formed from the two $2\times 2$ diagonal blocks ($D_1,D_2$) of a $4\times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{{\rho }_{11}{\rho }_{44}}{{\rho }_{22}{\rho }_{33}}}$--with, of central importance, $\mu =\epsilon$ or $\mu =\frac{1}{\epsilon }$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit "separability function" ${\tilde{\chi }}_1(\epsilon )$ ($\approx \epsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit (yielding $\frac{8}{33}$), two-quater[nionic]-bit (yielding $\frac{26}{323}$) and "two-octo[nionic]-bit" (yielding $\frac{44482}{4091349}$) counterparts. Then, we find a Lovas-Andai "master formula", $\widetilde{{\chi }_d}(\epsilon )=\frac{{\epsilon }^d\Gamma (d+1{)}^3{}_3{\tilde{F}}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3d}{2}+1;{\epsilon }^2\right)}{\Gamma {\left(\frac{d}{2}+1\right)}^2}$ encompassing both even and odd values of $d$. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence.

59 pages, 29 figures, retitled, added App. A, presenting previously-obtained (arXiv:0805.0267) analogous ABSOLUTE separability probabilities

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Cited by in corpus (4)

• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• On the generation of random ensembles of qubits and qutrits Computing separability probabilities for fixed rank states
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities