Mathematical Physics

   

Construction of the Lovas-Andai Two-Qubit Function $\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ Verifies the $\frac{8}{33}$-Hilbert Schmidt Separability Probability Conjecture

Authors: Paul B. Slater

We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit "separability functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ 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 randomly generated $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$--with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) 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 similarly obtain its new (much simpler) two-qubit counterpart, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to $\mu=\varepsilon$. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ($6 \times 6$) settings.

Comments: 35 pages, 26 figures

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Submission history

[v1] 2017-05-24 13:09:34

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