arXiv subjects

Paul B. Slater

Publications and source records attributed to Paul B. Slater.

At least 37 records · Page 2Linked to original sources

arXiv2016

Bloch Radii Repulsion in Separable Two-Qubit Systems

Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of $\frac{8}{33}$ for two-qubit systems ($ρ$) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius ($r$) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, $r=1$). Further, they also observed (personal communication) such separability probability $r$-invariance, when using, more broadly, random induced measure ($K=3,4,5,\ldots$), with $K=4$ corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality $|ρ^{PT}| >|ρ|$ and those from $|ρ| > |ρ^{PT}| >0$, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar $r$-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with $r$. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii ($r_A, r_B$) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) $X$-states model.

quant-ph↗

arXiv2015

Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Previously, a formula, incorporating a $5F4$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $\left\langle \left\vert ρ^{PT}\right\vert ^{n}\left\vert ρ\right\vert ^{k}\right\rangle /\left\langle \left\vert ρ\right\vert ^{k}\right\rangle$ of $4 \times 4$ density-matrices ($ρ$), and their partial transposes ($ρ^{PT}|$) was applied with $k=0$ to the generalized two-qubit separability-probability question. The formula can, further, be viewed we note here, as an averaging over "induced measures in the space of mixed quantum states". The associated induced-measure separability probabilities ($k =1, 2,\ldots$) are found--{\it via} a high-precision density approximation procedure--to assume interesting, relatively simple rational values in the two-re[al]bit ($α= \frac{1}{2}$), (standard) two-qubit ($α= 1$) and two-quater[nionic]bit ($α=2$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,\ldots) formulas that successfully reproduce the rational values assumed for {\it general} $k$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

quant-ph↗

arXiv2015

Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure

We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,α)= G_1^k(α) G_2^k(α)$, for $k = -1, 0, 1,\ldots,9$, for that (rational-valued) portion of the total separability probability for generalized two-qubit states endowed with random induced measure, for which the determinantal inequality $|ρ^{PT}| >|ρ|$ holds. Here $ρ$ denotes a $4 \times 4$ density matrix and $ρ^{PT}$, its partial transpose, while $α$ is a Dyson-index-like parameter with $α= 1$ for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is $4 \times (4 +k)$. For the symmetric case of $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard two-qubit case) $Q(0,1)=\frac{4}{33}$, and (the two-quater[nionic]bit case) $Q(0,2)= \frac{13}{323}$. The factors $G_2^k(α)$ can be written as the sum of weighted hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $\frac{27}{64} =(\frac{3}{4})^3$. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express $G_2^k(α)$ in terms of first-order difference equations. The factors $G_1^k(α)$ are equal to $(\frac{27}{64})^{α-1}$ times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable $α-$ and $k$-specific invariant asymptotic properties (again, involving $\frac{27}{64}$ and related quantities) of separability probability formulas emerge.

quant-ph↗

arXiv2015

Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures

Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert-Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general 2-qubit states--deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter $α$ employed.

quant-ph↗

arXiv2015

Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities

Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($ρ$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|ρ|>|ρ^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|ρ|=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {Ż}yczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.

quant-ph↗

arXiv2014

A Balanced $_{10}F_{9}$ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions

We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants $|ρ|$ and $|ρ^{PT}|$ of $4 \times 4$ density matrices, $ρ$, for generalized (rebit, quater[nionic]bit,\ldots) two-qubit systems, analogous to a series that has been obtained for the Hilbert-Schmidt (HS) metric. In particular, we desire--using moment-inversion procedures--to be able to closely test the previously-developed conjecture (J. Geom. Phys., 53, 74 [2005]) that the Bures separability probability over the (standard, fifteen-dimensional convex set of) two-qubit states is $\frac{1680 \left(\sqrt{2}-1\right)}{π^8} \approx 0.0733389$--while, in the HS context, strong evidence has been adduced, along the indicated analytical lines, that the counterpart of this value is $\frac{8}{33}$ (J. Phys. A}, 45, 095305 [2012]). Working within the "utility function" framework of Dunkl employed in that latter study, we obtain an interesting $_{10}F_{9}$ balanced hypergeometric function based on a "hybridization" of known Bures and HS terms. This exercise appears to provide an upper bound on the Bures two-qubit separability probability of 0.0798218. We also examine the yet unresolved HS qubit-qutrit scenario. Mathematica calculations indicate that if the same form of hypergeometric paradigm as has been established for the generalized two-qubit HS moments is followed in either the HS qubit-qutrit or Bures two-qubit cases, then a balanced hypergeometric function $_{p}F_{p-1}$ with $p>9$ would be required.

quant-ph↗

arXiv2014

Symmetric Halves of the 8/33-Probability that the Joint State of Two Quantum Bits is Disentangled

Compelling evidence-though yet no formal proof--has been adduced that the probability that a generic two-qubit state ($ρ$) is separable is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal moment formula of C. Dunkl (Appendix), we reach the conclusion that one-half of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|ρ|>|ρ^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The computational results reported lend strong support to those obtained earlier--including the "concise formula" that yields them--most conspicuously amongst those findings being the $\frac{29}{64}, \frac{8}{33}$ and $\frac{26}{323}$ probabilities noted.

quant-ph↗

arXiv2013

A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities

We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P(alpha) is put forth, yielding for generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and colleagues by applying Zeilberger's algorithm ("creative telescoping") to a fully equivalent--but considerably more complicated--expression containing six 7F6 hypergeometric functions (all with argument 27/64 = (3/4)^3). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7,501 determinantal moments of partially transposed four-by-four density matrices, parameterized by alpha = 1/2, 1, 3/2,...,32. From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) sequences of 32 rational values, then, served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P(alpha) emerged.

quant-ph↗

arXiv2013

Two-Qubit Rational-Valued Entanglement-Boundary Probability Densities and a Fisher Information Equality Conjecture

We consider a pair of one-parameter (alpha) families of generalized two-qubit determinantal Hilbert-Schmidt probability distributions, p_{alpha}(|rho^{PT}|) and q_{alpha}(|rho|), where rho is a 4 x 4 density matrix, rho^{PT}, its partial transpose, with |rho^{PT}| \in [-1/16,1/256] and |rho| \in [0, 1/256]. The Dyson-index-like (random matrix) parameter alpha is 1/2 for the 9-dimensional generic two-rebit systems, 1 for the 15-dimensional generic two-qubit systems,... Numerical (moment-based probability-distribution-reconstruction) analyses suggest the conjecture that the Fisher information--a measure over alpha--is identical for the two distinct families. Further, we study the values of p_{alpha}(0), the probability densities at the separability-entanglement boundary, with evidence strongly indicating that p_2(0) =7425/34 and p_3(0)= 7696/69. Despite extensive results of such a nature, we have not yet succeeded--in contrast with the corresponding rational-valued separability probabilities (arXiv.org:1301.6617)--in generating an underlying, explanatory ("concise") formula for p_α(0), even though the corresponding denominators of both sets of rational-valued results are closely related, having almost identical (small) prime factors. The first derivatives p_{alpha}^{'}(0) are positive for alpha = 1/2 and 1, but negative for alpha > 1.

quant-ph↗

arXiv2012

Toward Exact 2 x 2 Hilbert-Schmidt Determinantal Probability Distributions via Mellin Transforms and Other Approaches

We attempt to construct the exact univariate probability distributions for 2 x 2 quantum systems that yield the (balanced) univariate Hilbert-Schmidt determinantal moments <(|rho| |rho^{PT}|)^n>, obtained by Slater and Dunkl (J. Phys. A, 45, 095305 [2012]). To begin, we follow--to the extent possible--the Mellin transform-based approach of Penson and Zyczkowski in their study of Fuss-Catalan and Raney distributions (Phys. Rev. E, 83, 061118 [2011]). Further, we approximate the y-intercepts (separability/entanglement boundaries)--at which |rho^{PT}|=0-- of the associated probability distributions based on the (balanced) moments, as well as the previously reported unbalanced determinantal moments <|rho^{PT}|^n>, as a function of the seventy-two values of the Dyson-index-like parameter alpha = 1/2 (rebits), 1 (qubits),...,2 (quaterbits),...35.

quant-ph↗

arXiv2012

A Further (Itakura-Saito/beta=0) Bi-stochaticization and Associated Clustering/Regionalization of the 3,107-County 1995-2000 U. S. Migration Network

We extend to the beta-divergence (Itakura-Saito) case beta =0, the comparative bi-stochaticization analyses-previously conducted (arXiv:1208.3428) for the (Kullback-Leibler) beta=1 and (squared-Euclidean) beta = 2 cases -of the 3,107 - county 1995-2000 U. S. migration network. A heuristic, "greedy" algorithm is devised. While the largest 25,329 entries of the 735,531 non-zero entries of the bi-stochasticized table - in the beta=1 case - are required to complete the widely-applied two-stage (double-standardization and strong-component hierarchical clustering) procedure, 105,363 of the 735,531 are needed (reflective of greater uniformity of entries) in the beta=0 instance. The North Carolina counties of Mecklenburg (Charlotte) and Wake (Raleigh) are considerably relatively more cosmopolitan in the beta=0 study. The Colorado county of El Paso (Colorado Springs) replaces the Florida Atlantic county of Brevard (the "Space Coast") as the most cosmopolitan, with Brevard becoming the second-most. Honolulu County splinters away from the other four (still-grouped) Hawaiian counties, becoming the fifth most cosmopolitan county nation-wide. The five counties of Rhode Island remain intact as a regional entity, but the eight counties of Connecticut fragment, leaving only five counties clustered.

physics.soc-ph↗

arXiv2012

Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments

We seek to gain insight into the nature of the determinantal moments of generic (9-dimensional) two-rebit and (15-dimensional) two-qubit systems (rho). Such information-as it has proved to be in the Hilbert-Schmidt counterpart--should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining improved, or possibly even exact Bures 2 x 2 separability probabilities for such systems. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert-Schmidt moments (J. Phys. A, 45, 095305 [2012]). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures moments. In particular, the first moment (average) of |rho^{PT}| (where |rho^{PT}| in [-1/16, 1/256]), PT denoting partial transpose, for the two-qubit systems is, remarkably, -1 / 256. The analogous value for the two-rebit systems is -2663/(2^(13) x 3 x 5 x 7) approx -0.00309594. While, an important function in the Hilbert-Schmidt case is the ratio of 3n-degree polynomials, it appears to be the ratio of 5n-degree polynomials in the Bures case.

quant-ph↗

arXiv2012

Comparative Bi-stochastizations and Associated Clusterings/Regionalizations of the 1995-2000 U. S. Intercounty Migration Network

Wang, Li and Konig have recently compared the cluster-theoretic properties of bi-stochasticized symmetric data similarity (e. g. kernel) matrices, produced by minimizing two different forms of Bregman divergences. We extend their investigation to non-symmetric matrices, specifically studying the 1995-2000 U. S. 3,107 x 3,107 intercounty migration matrix. A particular bi-stochastized form of it had been obtained (arXiv:1207.0437), using the well-established Sinkhorn-Knopp (SK) (biproportional) algorithm--which minimizes the Kullback-Leibler form of the divergence. This matrix has but a single entry equal to (the maximal possible value of) 1. Highly contrastingly, the bi-stochastic matrix obtained here, implementing the Wang-Li-Konig-algorithm for the minimum of the alternative, squared-norm form of the divergence, has 2,707 such unit entries. The corresponding 3,107-vertex, 2,707-link directed graph has 2,352 strong components. These consist of 1,659 single/isolated counties, 654 doublets (thirty-one interstate in nature), 22 triplets (one being interstate), 13 quartets (one being interstate), three quintets and one septet. Not manifest in these graph-theoretic results, however, are the five-county states of Hawaii and Rhode Island and the eight-county state of Connecticut. These--among other regional configurations--appealingly emerged as well-defined entities in the SK-based strong-component hierarchical clustering.

cs.SI↗

arXiv2012

Two-Qubit Separability Probabilities: A Concise Formula

We report a concise answer--in the case of 2 x 2 systems--to the fundamental quantum-information-theoretic question as to "the volume of separable states" posed by Zyczkowski, Horodecki, Sanpera and Lewenstein (Phys. Rev. A, 58, 883 [1998]). We proceed by applying the Mathematica command FindSequenceFunction to a series of conjectured Hilbert-Schmidt generic 2 x 2 (rational-valued) separability probabilities p(a), a = 1, 2,...,32, with a = 1 indexing standard two-qubit systems, and a = 2, two-quater(nionic)bit systems. These 32 inputted values of p(a)--as well as 32 companion non-inputted values for the half-integers, a = 1/2 (two-re[al]bit) systems), 3/2,..., 63/2, are advanced on the basis of high-precision probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. The function P(a) given by application of the command fully reproduces both of these 32-length sequences, and an equivalent outcome is obtained if the half-integral series is the one inputted. The lengthy expression (containing six hypergeometric functions) obtained for P(a) is, then, impressively condensed (by Qing-Hu Hou and colleagues), using Zeilberger's algorithm. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33, (27-dimensional) two-quaterbit systems, P(2) = 26/323, while for generic classical (3-dimensional) systems, P(0)=1.

quant-ph↗

arXiv2012

A Hypergeometric Formula Yielding Hilbert-Schmidt Generic 2 x 2 Generalized Separability Probabilities

We significantly advance the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 =(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit, P(1) = 8/33, and (27-dimensional) two-quat(ernionic)bit systems, P(2)= 26/323. The function P(alpha) is generated--applying the Mathematica command FindSequenceFunction--using solely either) the thirty-two integral or half-integral probabilities, yet successfully predicts the unused thirty-two.

quant-ph↗

arXiv2012

Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities

Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n, k,n=0,1,2,3,..., PT denoting partial transpose, for both generic (9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit (alpha=1) density matrices rho. The results are, then, incorporated by Dunkl into a general formula (Appendix D6), parameterized by k, n and alpha, with the case alpha=2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the induced univariate moments (|rho| |rho^{PT}|)^n and |rho^{PT}|^n are inputted into a Legendre-polynomial-based (least-squares) probability-distribution reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding alpha-specific separability probability estimates. Since, as the number of inputted moments grows, estimates based on |rho| |rho^{PT}| strongly decrease, while ones employing |rho^{PT}| strongly increase (and converge faster), the gaps between upper and lower estimates diminish, yielding sharper and sharper bounds. Remarkably, for alpha = 2, with the use of 2,325 moments, a separability-probability lower-bound 0.999999987 as large as 26/323 = 0.0804954 is found. For alpha=1, based on 2,415 moments, a lower bound results that is 0.999997066 times as large as 8/33 = 0.242424, a (simpler still) fractional value that had previously been conjectured (J. Phys. A, 40, 14279 (2007)). Furthermore, for alpha = 1/2, employing 3,310 moments, the lower bound is 0.999955 times as large as 29/64 = 0.453125, a rational value previously considered (J. Phys. A, 43, 195302 (2010)).

quant-ph↗

arXiv2011

Radial and Azimuthal Profiles of Two-Qubit/Rebit Hilbert-Schmidt Separability Probabilities and Related 3-D Visualization Analyses

Firstly, we reduce the long-standing problem of ascertaining the Hilbert-Schmidt probability that a generic pair of qubits is separable to that of determining the specific nature of a one-dimensional (separability) function of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean space, and similarly for a generic pair of rebits, using the 9-dimensional unit ball. Separability probabilities, could, then, be directly obtained by integrating the products of these functions (which we numerically estimate and plot) with jacobian factors of r^m over r in [0,1], with m=17 for the two-rebit case, and m=29 in the two-qubit instance. Secondly, we repeat the analyses, but for the replacement of r as the free variable, by the azimuthal angle phi \in [0,2 pi]--with the associated jacobian factors now being, trivially, unity. So, the separability probabilities, then, become simply the areas under the curves generated. For our analyses, we employ an interesting Cholesky-decomposition parameterization of the 4 x 4 density matrices. Thirdly, in an exploratory investigation, we examine the Hilbert-Schmidt separability probability question using the three-dimensional visualization (cube/tetrahedron/octahedron) of two qubits associated with Avron, Bisker and Kenneth, and Leinaas, Myrheim and Ovrum, among others.

quant-ph↗

arXiv2011

Joint Hilbert-Schmidt Determinantal Moments of Product Form for the Two-Rebit and Two-Qubit and Higher-Dimensional Quantum Systems

We report formulas for the joint moments of the determinantal products (det{rho})^k (det{rho^PT})^K (k=0, 1, 2,...,N; K = 0, 1, 2, 3, 4) of Hilbert-Schmidt (HS) probability distributions over the two-rebit and (K = 0, 1) two-qubit density matrices rho. Here PT denotes the partial transposition operation of quantum-information-theoretic importance. Each formula is the product of the expression for the moments of (det{rho})^k, k=0,1,2,...,N--already known from work of Cappellini, Sommers and Zyczkowski (Phys. Rev. A 74, 062322 (1996))--and an adjustment factor. The factor is a biproper rational function, with its numerators and denominators both being 3 K-degree polynomials in k. We infer the structure that the denominators follow for arbitrary K in both the two-rebit and two-qubit cases, and the six leading-order coefficients of k of the numerators in the two-rebit scenario.. We also commence an analogous investigation of generic rebit-retrit and qubit-qutrit systems. This research was motivated, in part, by the objective of using the computed moments to well reconstruct the associated HS probabilities (for which we present plots) over the two determinants, and to ascertain--at least to high accuracy--the associated (separability) probabilities of "philosophical, practical and physical" interest that (det{rho^PT})>0.

quant-ph↗