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Paul B. Slater

Publications and source records attributed to Paul B. Slater.

At least 55 records · Page 3Linked to original sources
arXiv2011

Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

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.

quant-ph↗
arXiv2010

Multiscale Network Reduction Methodologies: Bistochastic and Disparity Filtering of Human Migration Flows between 3,000+ U. S. Counties

To control for multiscale effects in networks, one can transform the matrix of (in general) weighted, directed internodal flows to bistochastic (doubly-stochastic) form, using the iterative proportional fitting (Sinkhorn-Knopp) procedure, which alternatively scales row and column sums to all equal 1. The dominant entries in the bistochasticized table can then be employed for network reduction, using strong component hierarchical clustering. We illustrate various facets of this well-established, widely-applied two-stage algorithm with the 3, 107 x 3, 107 (asymmetric) 1995-2000 intercounty migration flow table for the United States. We compare the results obtained with ones using the disparity filter, for "extracting the "multiscale backbone of complex weighted networks", recently put forth by Serrano, Boguna and Vespignani (SBV) (Proc. Natl. Acad. Sci. 106 [2009], 6483), upon which we have briefly commented (Proc. Natl. Acad. Sci. 106 [2009], E66). The performance of the bistochastic filter appears to be superior-at least in this specific case-in two respects: (1) it requires far fewer links to complete a stongly-connected network backbone; and (2) it "belittles" small flows and nodes less-a principal desideratum of SBV-in the sense that the correlations of the nonzero raw flows are considerably weaker with the corresponding bistochastized links than with the significance levels yielded by the disparity filter. Additional comparative studies--as called for by SBV-of these two filtering procedures, in particular as regards their topological properties, should be of considerable interest. Relatedly, in its many geographic applications, the two-stage procedure has--with rare exceptions-clustered contiguous areas, often reconstructing traditional regions (islands, for example), even though no contiguity constraints, at all, are imposed beforehand.

physics.soc-ph↗
arXiv2010

Eigenvalues, Separability and Absolute Separability of Two-Qubit States

Substantial progress has recently been reported in the determination of the Hilbert-Schmidt (HS) separability probabilities for two-qubit and qubit-qutrit (real, complex and quaternionic) systems. An important theoretical concept employed has been that of a separability function. It appears that if one could analogously obtain separability functions parameterized by the eigenvalues of the density matrices in question--rather than the diagonal entries, as originally used--comparable progress could be achieved in obtaining separability probabilities based on the broad, interesting class of monotone metrics (the Bures, being its most prominent [minimal] member). Though large-scale numerical estimations of such eigenvalue-parameterized functions have been undertaken, it seems desirable also to study them in lower-dimensional specialized scenarios in which they can be exactly obtained. In this regard, we employ an Euler-angle parameterization of SO(4) derived by S. Cacciatori (reported in an Appendix)--in the manner of the SU(4)-density matrix parameterization of Tilma, Byrd and Sudarshan. We are, thus, able to find simple exact separability (inverse-sine-like) functions for two real two-qubit (rebit) systems, both having three free eigenvalues and one free Euler angle. We also employ the important Verstraete-Audenaert-de Moor bound to obtain exact HS probabilities that a generic two-qubit state is absolutely separable (that is, can not be entangled by unitary transformations). In this regard, we make copious use of trigonometric identities involving the tetrahedral dihedral angle arccos(1/3).

quant-ph↗
arXiv2010

Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes

The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.

quant-ph↗
arXiv2010

Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems

We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PT's) of the associated 4 x 4 density matrices). But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/{135 pi^2} =0.76854$. Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 x 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of 22/35 = 0.628571. Then, we conclude that a still further improved upper bound of 1129/2100 = 0.537619 can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 x 3 principal minors. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESF's that comes close to reproducing our numerical estimate of the true separability function.

quant-ph↗
arXiv2009

An improved exact upper bound of 22/35 on the Hilbert-Schmidt separability probability of real two-qubit systems

We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PTs) of the associated 4 x 4 density matrices. But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/(135 Pi^2) = 0.76854. Here, we extend this line of work by requiring that the four 3 x 3 principal minors of the PT be nonnegative, giving us an improved/reduced upper bound of 22/35 = 0.628571. Numerical simulations--as opposed to exact symbolic calculations--indicate, on the other hand, that the true probability is certainly less than 1/2. Our combined analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of 29/64 = 0.453125, while the best exact lower bound established so far is (6928-2205 Pi)/2^(9/2) = 0.0348338.

quant-ph↗
arXiv2009

Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions

The probability that a generic real, complex or quaternionic two-qubit state is separable can be considered to be the sum of three contributions. One is from those states that are absolutely separable, that is those (which can not be entangled by unitary transformations) for which the maximal concurrence over spectral orbits (C_{max}) is zero. The other two contributions are from the states for which C_{max} in [0,1/2], and for which C_{max} in [1/2,1]. We have previously (arXiv:0805.0267) found exact formulas for the absolutely separable contributions in terms of the Hilbert-Schmidt metric over the quantum states, and here advance hypotheses as to the exact contributions for C_{max} in [1/2,1]. A crucial element in understanding the other two contributions is the nature of the ratio (R) of the C_{max}-parameterized separability function for the complex states to the square of the comparable function for the real states--both such functions having clearly displayed jump discontinuities at C_{max}=1/2. For C_{max} in [0,1/2], R appears to be of the form 1+ k C_{max}, except near C_{max}=1/2, while for C_{max} in [1/2,1], it seems to be a constant (thus according to the Dyson-index pattern of random matrix theory). Related phenomena also occur for the minimally-degenerate two-qubit states and the qubit-qutrit states. Our results have immediate application to the computation of separability probabilities in terms of other metrics, such as the Bures (minimal monotone) metric. The paper begins with continuous embeddings of the separability probability question in terms of four metrics of interest, using "generalized Peres-Horodecki conditions".

quant-ph↗
arXiv2009

A two-stage algorithm for extracting the multiscale backbone of complex weighted networks

The central problem of concern to Serrano, Boguna and Vespignani ("Extracting the multiscale backbone of complex weighted networks", Proc Natl Acad Sci 106:6483-6488 [2009]) can be effectively and elegantly addressed using a well-established two-stage algorithm that has been applied to internal migration flows for numerous nations and several other forms of "transaction flow data".

physics.soc-ph↗
arXiv2009

Matrix plots of reordered bistochastized transaction flow tables: A United States intercounty migration example

We present a number of variously rearranged matrix plots of the $3, 107 \times 3, 107$ 1995-2000 (asymmetric) intercounty migration table for the United States, principally in its bistochasticized form (all 3,107 row and column sums iteratively proportionally fitted to equal 1). In one set of plots, the counties are seriated on the bases of the subdominant (left and right) eigenvectors of the bistochastic matrix. In another set, we use the ordering of counties in the dendrogram generated by the associated strong component hierarchical clustering. Interesting, diverse features of U. S. intercounty migration emerge--such as a contrast in centralized, hub-like (cosmopolitan/provincial) properties between cosmopolitan "Sunbelt" and provincial "Black Belt" counties. The methodologies employed should also be insightful for the many other diverse forms of interesting transaction flow-type data--interjournal citations being an obvious, much-studied example, where one might expect that the journals Science, Nature and PNAS would display "cosmopolitan" characteristics.

physics.soc-ph↗
arXiv2008

Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence. II--The Relevance of Dyson Indices

We importantly amend a certain parenthetical remark made in Part I (arXiv:0806.3294), to the effect that although two-qubit diagonal-entry-parameterized separability functions had been shown (arXiv:0704.3723) to clearly conform to a pattern dictated by the "Dyson indices" (beta = 1 [real], 2 [complex], 4 [quaternionic]) of random matrix theory, this did not appear to be the case with regard to eigenvalue-parameterized separability functions (ESFs). But upon further examination of the extensive numerical analyses reported in Part I, we find quite convincing evidence that adherence to the Dyson-index pattern does also hold for ESFs, at least as regards the upper half-range 1/2 <= C <= 1 of the maximal concurrence over spectral orbits, C. To be specific, it strongly appears that in this upper half-range, the real two-qubit ESF is simply proportional to (2-2 C)^(3/2), and its complex counterpart--in conformity to the Dyson-index pattern--proportional to the square of the real ESF, that is, (2-2 C)^3. The previously documented piecewise continuous ("semilinear") behavior in the lower half-range still appears, however, to lack any particular Dyson-index-related interpretation.

quant-ph↗
arXiv2008

Hubs and Clusters in the Evolving U. S. Internal Migration Network

Most nations of the world periodically publish N x N origin-destination tables, recording the number of people who lived in geographic subdivision i at time t and j at t+1. We have developed and widely applied to such national tables and other analogous (weighted, directed) socioeconomic networks, a two-stage--double-standardization and (strong component) hierarchical clustering--procedure. Previous applications of this methodology and related analytical issues are discussed. Its use is illustrated in a large-scale study, employing recorded United States internal migration flows between the 3,000+ county-level units of the nation for the periods 1965-1970 and 1995-2000. Prominent, important features--such as ''cosmopolitan hubs'' and ``functional regions''--are extracted from master dendrograms. The extent to which such characteristics have varied over the intervening thirty years is evaluated.

physics.soc-ph↗
arXiv2008

Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence

The generic real (b=1) and complex (b=2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert-Schmidt and Bures metrics are obtainable as special cases of formulas of Zyczkowski and Sommers. We claim that if one could determine certain metric-independent 3-dimensional "eigenvalue-parameterized separability functions" (EPSFs), then these formulas could be readily modified so as to yield the Hilbert-Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities). Motivated by analogous earlier analyses of "diagonal-entry-parameterized separability functions", we further explore the possibility that such 3-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable--which we hypothesize to be the maximal concurrence (0 < C <1) over spectral orbits. Extensive numerical results we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at C=1/2, as well as a number of additional matching discontinuities.

quant-ph↗
arXiv2008

Discernment of Hubs and Clusters in Socioeconomic Networks

Interest in the analysis of networks has grown rapidly in the new millennium. Consequently, we promote renewed attention to a certain methodological approach introduced in 1974. Over the succeeding decade, this two-stage--double-standardization and hierarchical clustering (single-linkage-like)--procedure was applied to a wide variety of weighted, directed networks of a socioeconomic nature, frequently revealing the presence of ``hubs''. These were, typically--in the numerous instances studied of migration flows between geographic subdivisions within nations--``cosmopolitan/non-provincial'' areas, a prototypical example being the French capital, Paris. Such locations emit and absorb people broadly across their respective nations. Additionally, the two-stage procedure--which ``might very well be the most successful application of cluster analysis'' (R. C. Dubes, 1985)--detected many (physically or socially) isolated, functional groups (regions) of areas, such as the southern islands, Shikoku and Kyushu, of Japan, the Italian islands of Sardinia and Sicily, and the New England region of the United States. Further, we discuss a (complementary) approach developed in 1976, in which the max-flow/min-cut theorem was applied to raw/non-standardized (interindustry, as well as migration) flows.

physics.soc-ph↗
arXiv2008

Established Clustering Procedures for Network Analysis

In light of the burgeoning interest in network analysis in the new millenium, we bring to the attention of contemporary network theorists, a two-stage double-standarization and hierarchical clustering (single-linkage-like) procedure devised in 1974. In its many applications over the next decade--primarily to the migration flows between geographic subdivisions within nations--the presence was often revealed of ``hubs''. These are, typically, ``cosmopolitan/non-provincial'' areas--such as the French capital, Paris--which send and receive people relatively broadly across their respective nations. Additionally, this two-stage procedure--which ``might very well be the most successful application of cluster analysis'' (R. C. Dubes)--has detected many (physically or socially) isolated groups (regions) of areas, such as those forming the southern islands, Shikoku and Kyushu, of Japan, the Italian islands of Sardinia and Sicily, and the New England region of the United States. Further, we discuss a (complementary) approach developed in 1976, involving the application of the max-flow/min-cut theorem to raw/non-standardized flows.

physics.soc-ph↗
arXiv2008

Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices

We report substantial progress in the study of separability functions and their application to the computation of separability probabilities for the real, complex and quaternionic qubit-qubit and qubit-qutrit systems. We expand our recent work (arXiv:0704.3723), in which the Dyson indices of random matrix theory played an essential role, to include the use of not only the volume element of the Hilbert-Schmidt (HS) metric, but also that of the Bures (minimal monotone) metric as measures over these finite-dimensional quantum systems. Further, we now employ the Euler-angle parameterization of density matrices (rho), in addition to the Bloore parameterization. The Euler-angle separability function for the minimally degenerate complex two-qubit states is well-fitted by the sixth-power of the participation ratio, R(rho)=1/Tr(rho)^2. Additionally, replacing R(rho) by a simple linear transformation of the Verstraete-Audenaert-De Moor function (arXiv:quant-oh/0011111), we find close adherence to Dyson-index behavior for the real and complex (nondegenerate) two-qubit scenarios. Several of the analyses reported help to fortify our conjectures that the HS and Bures separability probabilities of the complex two-qubit states are 8/33 = 0.242424 and 1680 (sqrt{2}-1)/pi^8 = 0.733389, respectively. Employing certain regularized beta functions in the role of Euler-angle separability functions, we closely reproduce--consistently with the Dyson-index ansatz--several HS two-qubit separability probability conjectures.

quant-ph↗
arXiv2007

Two-Qubit Hilbert-Schmidt Separability Functions and Probabilities for Full-Dimensional Even-Dyson-Index Scenarios

We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional convex set of two-qubit complex (beta=2) systems, as well as the fourth power of the separability function for the 9-dimensional convex set of two-qubit real (beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2 ((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22} rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus, supplement (and fortify) our previous assertion that the HS separability probability of the two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze the odd beta=1 and 3 cases.

quant-ph↗