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.
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.
A complete description of the multitudinous ways in which quantum particles can be entangled requires the use of high-dimensional abstract mathematical spaces. We report here a particularly interesting feature of the nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat) measure-composed of two-rebit (4 x 4) density matrices (rho). To each rho is assigned the product of its (nonnegative) determinant |rho| and the determinant of its partial transpose |rho^{PT}| -negative values of which, by the results of Peres and Horodecki, signify the entanglement of rho. Integrating this product, |rho| |rho^{PT}| =|rho rho^{PT}|, over the nine-dimensional space, using the indicated (HS) measure, we obtain the result zero. The two determinants, thus, form a pair of multivariate orthogonal polynomials with respect to HS measure. It is hypothesized that the analogous two determinants for the standard fifteen-dimensional convex set of two-qubit density matrices are similarly orthogonal. However, orthogonality does not hold, we find, if the symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor with the use of non-flat measures, such as the prominent Bures (minimal monotone) measure-nor in the full HS extension to the 15-dimensional convex set of two-qubit density matrices. We discuss relations-involving the HS moments of |ρ^{PT}|-to the long-standing problem of determining the probability that a generic pair of rebits/qubits is separable.
We seek to gain insight into the nature of the determinantal moments 〈|ρ PT | n |ρ| k 〉 Bures of generic (nine-dimensional) two-rebit and (fifteen-dimensional) two-qubit systems (ρ), PT denoting partial transpose. Such information—as it has proved to be in the Hilbert–Schmidt counterpart—should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining Bures 2 × 2 separability probabilities. 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 (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures determinantal moments. In particular, the first moment (average) of |ρ PT | (where ) for the two-qubit systems is, remarkably, . The analogous value for the two-rebit systems is . While in the Hilbert–Schmidt case is the ratio of 3 n -degree polynomials in k , it appears to be the ratio of 5 n -degree polynomials in k in the Bures case.
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)).
We attempt to construct the exact univariate probability distributions for 2 x 2 quantum systems that yield the (balanced) univariate Hilbert-Schmidt determinantal moments , 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 , as a function of the seventy-two values of the Dyson-index-like parameter alpha = 1/2 (rebits), 1 (qubits),...,2 (quaterbits),...35.
Previously, a formula, incorporating a 5F4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments ρPTnρk/ρk of 4×4 density-matrices ( ρ ) and their partial transposes ( |ρPT| ), was applied with k=0 to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as 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,… ) are found—viaa high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit ( α=1/2 ), (standard) two-qubit ( α=1 ), and two-quater[nionic]bit ( α=2 ) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed forgeneral k . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master 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.