Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities

Abstract

Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |ρ| k |ρ P T | n , k, n = 0, 1, 2, 3, . . ., P T denoting partial transpose, for both generic (9-dimensional) two-rebit (α = 1 2 ) and generic (15-dimensional) two-qubit (α = 1) density matrices ρ. The results are, then, incorporated by Dunkl into a general formula (Appendix D 6), parameterized by k, n and α, with the case α = 2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter α fixed, the induced univariate moments (|ρ||ρ P T |) n and |ρ P T | n are inputted into a Legendre-polynomialbased (least-squares) probability-distribution reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding α-specific separability probability estimates. Since, as the number of inputted moments grows, estimates based on the variable |ρ||ρ P T | strongly decrease, while ones employing |ρ P T | strongly increase (and converge faster), the gaps between upper and lower estimates diminish, yielding sharper and sharper bounds. Remarkably, for α = 2, with the use of 2,325 moments, a separability-probability lower-bound 0.999999987 as large as 43, 195302 (2010)).

Invariance of bipartite separability and PPT-probabilities over Casimir invariants of reduced states

“…[40]). Proofs are certainly still lacking for the interesting generalized Bloch radii/Casimir invariants conjectures made above, as well as those in a number of other recent reports [11][12][13][14]20,22,36] and the infinite summation formula given above (1)(2)(3).…”

“…In regard to these matters, a diverse body of evidence-though yet no formal proofhas been developed, strongly indicating that the probability that a two-qubit state is separable/disentangled/classically correlated, that is, expressible as the convex sum of products of qubit states [10], is 8 33 ≈ 0.242424 [11][12][13][14]. The measure employed in the underlying computations was the familiar Hilbert-Schmidt (Euclidean/flat) one [2,15], while the integration of this measure was conducted over the standard 15-dimensional convex set of 4 × 4 (Hermitian) density matrices.…”

Extensions of generalized two-qubit separability probability analyses to higher dimensions, additional measures and new methodologies

“…This decomposes into the sum of the Lovas-Andai separability functions 1 20 ε 2 (ε 4 − 6ε 2 + 15) and and qubit-qudit states, among others, examined above. But, given that the Hilbert-Schmidt separability probability of 8 33 for the 15-dimensional convex set of two-qubit states has itself proved highly formidable to well establish [1,[4][5][6][7][8][9][10] (though not yet formally proven), it seems that major advances would be required to achieve such a goal in these still higher-dimensional settings (and, thus, confirm or reject the conjectures above). Implicit in the analytical approach pursued here has been the clearly yet unverified assumption that the separability/PPT-probabilities will continue to be rational-valued for the higher-dimensional systems, as they have, remarkably, been found to be in the 4 × 4 setting.…”

“…One of our goals here has been to determine if we could use the N = 4 results [1,[4][5][6][7][8][9][10] to gain insight into the N > 4 counterparts, and, more specifically, if certain analytical properties continue to hold. We found some encouragement for undertaking such a course from the research reported in [30].…”

Formulas for Generalized Two-Qubit Separability Probabilities

“…To obtain the new formulas ( , ) to be presented here for the separability probability amounts for which | PT | > | | holds, we first employed-as in our prior studies [1,9,16,17]-the Legendre-polynomial-based probability density-approximation (Mathematica-implemented) algorithm of Provost [18] (cf. [19]).…”