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-rebit 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 ε ∈ [0, 1] is the ratio σ(V ) of the singular values offormed from the two 2 × 2 diagonal blocks (D 1 , D 2 ) of a 4 × 4 density matrix D = ρ ij . In the Slater setting, the independent variable µ is the diagonal-entry ratio ρ 11 ρ 44 ρ 22 ρ 33 -with, of central importance, µ = ε or µ = 1 ε when both D 1 and D 2 are themselves diagonal. Lovas and Andai established that their two-rebit "separability function" χ1 (ε) (≈ ε) yields the previously conjectured Hilbert-Schmidt separability probability of 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, χ2 (εThese immediately lead to predictions of Hilbert-Schmidt separability/PPT-probabilities of 8 33 , 26 323 and 44482 4091349 , in full agreement with those of the "concise formula" (J. Phys. A 46 [2013] 445302), and, additionally, of a "specialized induced measure" formula. Then, we find a Lovas-Andai "master formula", χd (ε) =+1) 2 , encompassing both even and odd values of d. Remarkably, we are able to obtain the χd (ε) 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 − 256 27π 2 is obtained based on the operator monotone function √ x, with the use of χ2 (ε).

On the Generation of Random Ensembles of Qubits and Qutrits Computing Separability Probabilities for Fixed Rank States

“…The first method is suitable for analytic calculations, however it presents a big computational issue (see e.g., [2] and references therein). The second method allows us to use highly effective modern numerical computational techniques.…”