We begin by investigating relationships between two forms of Hilbert–Schmidt two-rebit and two-qubit “separability functions”—those recently advanced by Lovas and Andai (J Phys A Math Theor 50(29):295303,
2017), and those earlier presented by Slater (J Phys A 40(47):14279,
2007). 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
\(4 \times 4\) density matrix
\(D= \left||\rho _{ij}\right||\). 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,
\(\mu =\varepsilon \) or
\(\mu =\frac{1}{\varepsilon }\) when both
\(D_1\) and
\(D_2\) are themselves diagonal. Lovas and Andai established that their two-rebit “separability 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 newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts,
\(\tilde{\chi _2}(\varepsilon ) =\frac{1}{3} \varepsilon ^2 \left( 4-\varepsilon ^2\right) \),
\(\tilde{\chi _4}(\varepsilon ) =\frac{1}{35} \varepsilon ^4 \left( 15 \varepsilon ^4-64 \varepsilon ^2+84\right) \) and
\(\tilde{\chi _8} (\varepsilon )= \frac{1}{1287}\varepsilon ^8 \left( 1155 \varepsilon ^8-7680 \varepsilon ^6+20160 \varepsilon ^4-25088 \varepsilon ^2+12740\right) \). These immediately lead to predictions of Hilbert–Schmidt separability/PPT-probabilities of
\(\frac{8}{33}\),
\(\frac{26}{323}\) and
\(\frac{44482}{4091349}\), in full agreement with those of the “concise formula” (Slater in J Phys A 46:445302,
2013), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas–Andai “master formula,”
\(\tilde{\chi _d}(\varepsilon )= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left( -\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right) }{\Gamma \left( \frac{d}{2}+1\right) ^2}\), encompassing both even and odd values of
d. Remarkably, we are able to obtain the
\(\tilde{\chi _d}(\varepsilon )\) formulas,
\(d=1,2,4\), applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal
\(D_1\) and
\(D_2\), but also an additional pair of nullified entries. Nullification of a further pair still leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of
\(1-\frac{256}{27 \pi ^2}\) is obtained based on the
operator monotone function
\(\sqrt{x}\), with the use of
\(\tilde{\chi _2}(\varepsilon )\).
