arXiv subjects

Paul B. Slater

Publications and source records attributed to Paul B. Slater.

At least 19 recordsLinked to original sources

arXiv2021

Density Matrix Diagonal-Block Lovas-Andai-type singular-value ratios for qubit-qudit separability/PPT probability analyses

An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of $\frac{29}{64}$ for the 9-dimensional convex set of two-rebit density matrices, was the ratio ($\varepsilon =\frac{σ_2}{σ_1}$) of the two singular values ($σ_1 \geq σ_2 \geq 0$) of $D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$. There, $D_1$ and $D_2$ were the diagonal $2 \times 2$ blocks of a $4 \times 4$ two-rebit density matrix $ρ$. Working within the Lovas-Andai "separability function" ($\tildeχ_d(\varepsilon)$) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of $\frac{8}{33}$ and $\frac{26}{323}$ for the 15-dimensional and 26-dimensional convex sets of two-qubit and two-quater[nionic]-bit density matrices. Here, we investigate the behavior of the three singular value ratios of $V=D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$, where now $D_1$ and $D_2$ are the $3 \times 3$ diagonal blocks of $6 \times 6$ rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing $8 \times 8$ density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending $\tildeχ_d(\varepsilon)$ to higher dimensional systems--an issue we also approach using certain novel numeric means. Section 3.3 of the 2017 A. Lovas doctoral dissertation (written in Hungarian) appears germane to such an investigation.

quant-ph↗

arXiv2021

Rational-Valued, Small-Prime-Based Qubit-Qutrit and Rebit-Retrit Rank-4/Rank-6 Conjectured Hilbert-Schmidt Separability Probability Ratios

We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) $N \times N$ density matrices of rank $k \leq N$ with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix $A$ of dimensions $k \times k+ 2 (N-k)$, while for a real scenario, one employs a Ginibre matrix $B$ of dimensions $k \times k+1+ 2 (N-k)$. Then, the $k \times k$ product $A A^{\dagger}$ or $B B^T$ is diagonalized-padded with zeros to size $N \times N$-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value $\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^4}=.0774$. A conjecture for the HS separability probability of rebit-retrit systems of full rank is $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775$ (the two-rebit counterpart has been proven to be $\frac{29}{64}=\frac{29}{2^6}$). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be $\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049$, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {Ż}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be $\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3}$, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a $\frac{70}{2673}=\frac{2 \cdot 5 \cdot 7}{ 3^5 \cdot 11} \approx 0.0261878$ rank-4 to rank-6 probability ratio.

quant-ph↗

arXiv2021

Hilbert-Schmidt Separability Probabilities from Bures Ensembles and vice versa: Applications to Quantum Steering Ellipsoids and Monotone Metrics

We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $Π_{j<k}^{1...4} (λ_j-λ_k)^2 $ term of the HS volume element (the $λ$'s being the four eigenvalues of the associated $4 \times 4$ density matrix $ρ$). This measure is applied to the members of a HS ensemble of random two-qubit states, yielding a QES separability probability estimate of 0.105458. Alternatively, weighting members of a Bures ensemble by the QES volume divided by the eigenvalue part $\frac{1}{\sqrt{\mbox{det} ρ}} Π_{j<k}^{1...4} \frac{(λ_j-λ_k)^2}{λ_j+λ_k}$ of the Bures volume element, gives a close estimate of 0.100223. We also weight members of a HS ensemble by the QES volume divided not only by the indicated HS eigenvalue term, but also by the unitary component $|Π_{j<k}^{1...4} \mbox{Re} (U^{-1}) \mbox{Im} (U^{-1})|$ of the volume element. For one hundred thirty (rather variable) independent separability probability estimates, we, then, obtain median and mean estimates of 0.0447729 and 0.117485 with variance 0.0381468.

quant-ph↗

arXiv2020

Quantum Steering Ellipsoid Volume as a Measure on the Two-Qubit States and Associated Separability and Absolute Separability Ratios

Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.

quant-ph↗

arXiv2020

Archipelagos of Total Bound and Free Entanglement. II

In the indicated preceding preprint (I), we reported the results of, in particular interest here, certain three-parameter qubit-ququart ($2 \times 4$) and two-ququart ($4 \times 4$) analyses. In them, we relied upon entanglement constraints given by Li and Qiao. However, further studies of ours conclusively show--using the well-known necessary and sufficient conditions for positive-semidefiniteness that all leading minors (of separable components, in this context) be nonnegative--that certain of the constraints given are flawed and need to be replaced (by weaker ones). Doing so, leads to a new set of results, somewhat qualitatively different and, in certain respects, simpler in nature. For example, bound-entanglement probabilities of $\frac{2}{3} \left(\sqrt{2}-1\right) \approx 0.276142$, $\frac{1}{4} \left(3-2 \log ^2(2)-\log (4)\right) \approx 0.1632$, $\frac{1}{2}-\frac{2}{3 π^2} \approx 0.432453$ and $\frac{1}{6}$, are reported for various implementations of constraints. We also adopt the Li-Qiao three-parameter framework to a two-parameter one, with interesting visual results.

quant-ph↗

arXiv2020

8 Boolean Atoms Spanning the 256-Dimensional Entanglement-Probability Three-Set Algebra of the Two-Qutrit Hiesmayr-Loffler Magic Simplex of Bell States

We obtain formulas (bot. p. 12)--including $\frac{2}{121}$ and $\frac{4 \left(242 \sqrt{3} π-1311\right)}{9801}$--for the eight atoms (Fig. 11), summing to 1, which span a 256-dimensional three-set (P, S, PPT) entanglement-probability boolean algebra for the two-qutrit Hiesmayr-Loffler states. PPT denotes positive partial transpose, while P and S provide the Li-Qiao necessary and}sufficient conditions for entanglement. The constraints ensuring entanglement are $s> \frac{16}{9} \approx 1.7777$ and $p> \frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}$. Here, $s$ is the square of the sum (Ky Fan norm) of the eight singular values of the $8 \times 8$ correlation matrix in the Bloch representation, and $p$, the square of the product of the singular values. In the two-ququart Hiesmayr-Loffler case, one constraint is $s>\frac{9}{4} \approx 2.25$, while $\frac{3^{24}}{2^{134}} \approx 1.2968528306 \cdot 10^{-29}$ is an upper bound on the appropriate $p$ value, with an entanglement probability $\approx 0.607698$. The $S$ constraints, in both cases, prove equivalent to the well-known CCNR/realignment criteria. Further, we detect and verify--using software of A. Mandilara--pseudo-one-copy undistillable (POCU) negative partial transposed two-qutrit states distributed over the surface of the separable states. Additionally, we study the best separable approximation problem within this two-qutrit setting, and obtain explicit decompositions of separable states into the sum of eleven product states. Numerous quantities of interest--including the eight atoms--were, first, estimated using a quasirandom procedure.

quant-ph↗

arXiv2020

Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

We report several witness-parameterized families of bound-entangled probabilities. Two pertain to the $d=3$ (two-qutrit) and a third to the $d=4$ (two-ququart) subsets analyzed by Hiesmayr and L{ö}ffler of "magic" simplices of Bell states. The Hilbert-Schmidt probabilities of positive-partial-transpose (PPT) states--within which we search for bound-entangled states--are $\frac{8 π}{27 \sqrt{3}} \approx 0.537422$ ($d=3$) and $\frac{1}{2}+\frac{\log \left(2-\sqrt{3}\right)}{8 \sqrt{3}} \approx 0.404957$ ($d=4$). We obtain bound-entangled probabilities of $-\frac{4}{9}+\frac{4 π}{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$ and $\frac{-204+7 \log (7)+168 \sqrt{3} \cos ^{-1}\left(\frac{11}{14}\right)}{1134} \approx 0.00325613$ ($d=3$) and $\frac{8 \log (2)}{27}-\frac{59}{288} \approx 0.00051583$ and $\frac{24 \text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17 \sqrt{17}}-\frac{91}{544} \approx 0.00218722$ ($d=4$). Thus, the total entanglement probability appears to equal $(1-\frac{8 π}{27 \sqrt{3}})+\frac{2}{81} \left(4 \sqrt{3} π-21\right) = \frac{13}{27} \approx 0.481481$.) The families, encompassing these results, are parameterized using generalized Choi and Jafarizadeh-Behzadi-Akbari witnesses. The same bound-entangled probability was achieved with both--the sets ("jagged islands") detected having void intersection. The entanglement (bound and "non-bound"/"free") probability for both was $\frac{1}{6} \approx 0.16667$, while their union and intersection gave $\frac{2}{9} \approx 0.22222$ and $\frac{1}{9} \approx 0.11111$. Further, we examine generalized Horodecki states, as well as estimating PPT-probabilities of approximately 0.39339 (very well-fitted by $\frac{7 π}{25 \sqrt{5}} \approx 0.39338962$) and 0.115732 ( for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.

quant-ph↗

arXiv2020

Quasirandom estimations of two-qubit operator-monotone-based separability probabilities

We conduct a pair of quasirandom estimations of the separability probabilities with respect to ten measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parameterization. The measures include the (non-monotone) Hilbert-Schmidt one, plus nine others based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert-Schmidt and Bures (minimal monotone) separability probabilities are $\frac{8}{33} \approx 0.242424$ and $\frac{25}{341} \approx 0.0733138$, respectively, as well as suggestive of the Wigner-Yanase counterpart being $\frac{1}{20}$. However, one result appears inconsistent (much too small) with an earlier claim of ours that the separability probability associated with the operator monotone (geometric-mean) function $\sqrt{x}$ is $1-\frac{256}{27 π^2} \approx 0.0393251$. But a seeming explanation for this disparity is that the volume of states for the $\sqrt{x}$-based measure is infinite. So, the validity of the earlier conjecture--as well as an alternative one, $\frac{1}{9} \left(593-60 π^2\right) \approx 0.0915262$, we now introduce--can not be examined through the numerical approach adopted, at least perhaps not without some truncation procedure for extreme values.

quant-ph↗

arXiv2020

Archipelagos of Total Bound and Free Entanglement

First, we considerably simplify an initially quite complicated formula -- involving dilogarithms. It yields the total bound entanglement probability ($\approx 0.0865542$) for a qubit-ququart ($2 \times 4$) three-parameter model, recently analyzed for its separability properties by Li and Qiao. An "archipelago" of disjoint bound-entangled regions appears in the space of parameters, somewhat similarly to those recently found in our preprint, "Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities". There, two-qutrit and two-ququart Hiesmayr-L{ö}ffler "magic simplices" and generalized Horodecki states had been examined. However, contrastingly, in the present study, the entirety of bound entanglement--given by the formula obtained--is clearly captured in the archipelago found. Further, we "upgrade" the qubit-ququart model to a two-ququart one, for which we again find a bound-entangled archipelago, with its total probability simply being now $\frac{1}{729} \left(473-512 \log \left(\frac{27}{16}\right) \left(1+\log \left(\frac{27}{16}\right)\right)\right) \approx 0.0890496$. Then, "downgrading" the qubit-ququart model to a two-qubit one, we find an archipelago of total non-bound/free entanglement probability $\frac{1}{2}$.

quant-ph↗

arXiv2019

Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities

We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion iterations, we do not presently perceive an exact value (rational or otherwise) for an estimate of 0.15709623 for the Bures two-rebit separability probability. The Bures qubit-qutrit case--for which Khvedelidze and Rogojin gave an estimate of 0.0014--is analyzed too. The value of $\frac{1}{715}=\frac{1}{5 \cdot 11 \cdot 13} \approx 0.00139860$ is a well-fitting value to an estimate of 0.00139884. Interesting values ($\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11}$ and $\frac{625}{109531136}=\frac{5^4}{2^{12} \cdot 11^2 \cdot 13 \cdot 17}$) are conjectured for the Hilbert-Schmidt (HS) and Bures qubit-qudit ($2 \times 4$) positive-partial-transpose (PPT)-probabilities. We re-examine, strongly supporting, conjectures that the HS qubit-{\it qutrit} and rebit-{\it retrit} separability probabilities are $\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively. Prior studies have demonstrated that the HS two-rebit separability probability is $\frac{29}{64}$ and strongly pointed to the HS two-qubit counterpart being $\frac{8}{33}$, and a certain operator monotone one (other than the Bures) being $1 -\frac{256}{27 π^2}$.

quant-ph↗

arXiv2019

Ordinal and Cardinal Dendrograms Depicting Migration-Based Regionalization of 3,000 + U. S. Counties

We have obtained a "hierarchical regionalization" of 3,107 county-level units of the United States based upon census-recorded 1995-2000 intercounty migration flows. The methodology employed was the two-stage (double-standardization and strong component [directed graph] hierarchical clustering) algorithm described in the 2009 PNAS (106 [26], E66) letter (arXiv:0904.4863). Various features (e. g., cosmopolitan vs. provincial aspects, and indices of isolation) of the regionalization have been previously discussed in arXiv:0907.2393, arXiv:0903.3623 and arXiv:0809.2768. However, due to the lengthy (38-page) nature of the associated dendrogram, the detailed tree structure itself was not readily available for inspection. Here, we do present this (county-searchable) dendrogram--and invite readers to explore it, based on their particular interests/locations. An ordinal scale--rather than the originally-derived cardinal scale of the doubly-standardized values--in which groupings/features were more immediately apparent, was originally presented. Now, we append the cardinal-scale dendrogram.

physics.soc-ph↗

arXiv2018

Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies

We first seek the rebit-retrit counterpart to the (formally proven by Lovas and Andai) two-rebit Hilbert-Schmidt separability probability of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$ and the qubit-qutrit analogue of the (strongly supported) value of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$. We advance the possibilities of a rebit-retrit value of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$ and a qubit-qutrit one of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$. These four values for $2 \times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 π^2} \approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets. Then, we extend the master Lovas-Andai formula to induced measures. For instance, the two-qubit function ($k=0$) is $\tildeχ_{2,0}(\varepsilon)=\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, while its $k=1$ induced measure counterpart is $\tildeχ_{2,1}(\varepsilon)=\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. Interpolations between Hilbert-Schmidt and operator monotone (Bures, $\sqrt{x}$) measures are also studied. Using a recently-developed golden-ratio-related (quasirandom sequence) approach, current (significant digits) estimates of the two-rebit and two-qubit Bures separability probabilities are 0.15709 and 0.07331, respectively.

quant-ph↗

arXiv2018

Qubit-Qudit Separability/PPT-Probability Analyses and Lovas-Andai Formula Extensions to Induced Measures

We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$. These four values for $2 \times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 π^2} \approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets of $X$-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is $\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, its $k=1$ induced measure counterpart is $\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".

quant-ph↗

arXiv2017

Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures

We begin by investigating relationships between two forms of Hilbert-Schmidt two-re[al]bit 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 $\varepsilon \in [0,1]$ is the ratio $σ(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $μ$ is the diagonal-entry ratio $\sqrt{\frac{ρ_{11} ρ_{44}}{ρ_{22} ρ_{33}}}$--with, of central importance, $μ=\varepsilon$ or $μ=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit "separability function" $\tildeχ_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. 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 (yielding $\frac{8}{33}$), two-quater[nionic]-bit (yielding $\frac{26}{323}$) and "two-octo[nionic]-bit" (yielding $\frac{44482}{4091349}$) counterparts. Then, we find a Lovas-Andai "master formula", $\tilde{χ_d}(\varepsilon)= \frac{\varepsilon ^d Γ(d+1)^3 \,_3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3d}{2}+1;\varepsilon ^2\right)}{Γ\left(\frac{d}{2}+1\right)^2}$ encompassing both even and odd values of $d$. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence.

quant-ph↗

arXiv2017

Formulas for Generalized Two-Qubit Separability Probabilities

To begin, we find certain formulas $Q(k,α)= G_1^k(α) G_2^k(α)$, for $k = -1, 0, 1,...,9$. These yield that part of the total separability probability, $P(k,α)$, for generalized (real, complex, quaternionic,\ldots) 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, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $ρ^{PT}$, its partial transpose. Further, $α$ is a Dyson-index-like parameter with $α= 1$ for the standard (15-dimensional) convex set of (complex) two-qubit states. For $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the real case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard complex case) $Q(0,1)=\frac{4}{33}$, and (the quaternionic case) $Q(0,2)= \frac{13}{323}$---the three simply equalling $ P(0,α)/2$. The factors $G_2^k(α)$ are sums of polynomial-weighted generalized hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $z=\frac{27}{64} =(\frac{3}{4})^3$. We find number-theoretic-based formulas for the upper ($u_{ik}$) and lower ($b_{ik}$) parameter sets of these functions and, then, equivalently express $G_2^k(α)$ in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for $P(0,α) =2 Q(0,α)$. For nonnegative half-integer and integer values of $α$, $Q(k,α)$ has descending roots starting at $k=-α-1$. Then, we (C. Dunkl and I) construct a remarkably compact (hypergeometric) form for $Q(k,α)$ itself. The possibility of an analogous "master" formula for $P(k,α)$ is, then, investigated, and a number of interesting results found.

quant-ph↗

arXiv2016

Octonionic two-qubit separability probability conjectures

We study, further, a conjectured formula for generalized two-qubit Hilbert-Schmidt separability probabilities that has recently been proven by Lovas and Andai (https://arxiv.org/pdf/1610.01410.pdf) for its real (two-rebit) asserted value ($\frac{29}{64}$), and that has also been very strongly supported numerically for its complex ($\frac{8}{33}$), and quaternionic ($\frac{26}{323}$) counterparts. Now, we seek to test the presumptive octonionic value of $\frac{44482}{4091349} \approx 0.0108722$. We are somewhat encouraged by certain numerical computations, indicating that this (51-dimensional) instance of the conjecture might be fulfilled by setting a certain determinantal-power parameter $a$, introduced by Forrester (https://arxiv.org/pdf/1610.08081.pdf), to 0 (or possibly near to 0). Hilbert-Schmidt measure being the case $k=0$ of random induced measure, for $k=1$, the corresponding octonionic separability probability conjecture is $\frac{7612846}{293213345} \approx 0.0259635$, while for $k=2$, it is $\frac{4893392}{95041567} \approx 0.0514869, \ldots$. The relation between the parameters $a$ and $k$ is explored.

quant-ph↗

arXiv2016

Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems

We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert-Schmidt, and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0 \leq r_A,r_B \leq 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A) < p(r_A,1-r_A)$, except for rather narrow "crossover" intervals $[\tilde{r}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the "toy" seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in $4 \times K$-dimensions, for $K=3, 4, 5$, with $K=4$ corresponding to Hilbert-Schmidt (flat/Euclidean) measure. We also examine the real (two-rebit) $K=4$, the $X$-states $K=5$, and Bures (minimal monotone)--for which no nontrivial crossover behavior is observed--instances. In the two $X$-states cases, we derive analytical results, for $K=3, 4$, we propose formulas that well-fit our numerical results, and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower $\tilde{r}_A$) as $K$ increases. This report continues our efforts (arXiv:1506.08739) to extend the recent work of Milz and Strunz (J. Phys. A: 48 [2015] 035306) from a univariate ($r_A$) framework---in which they found separability probabilities to hold constant with $r_A$---to a bivariate ($r_A,r_B$) one. We also analyze the two-qutrit and qubit-qutrit counterparts reported in arXiv:1512.07210 in this context, and study two-qubit separability probabilities of the form $p(r_A,\frac{1}{2})$.

quant-ph↗

arXiv2016

Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States

Milz and Strunz ({\it J. Phys. A}: {\bf{48}} [2015] 035306) recently studied the probabilities that two-qubit and qubit-qutrit states, randomly generated with respect to Hilbert-Schmidt (Euclidean/flat) measure, are separable. They concluded that in both cases, the separability probabilities (apparently exactly $\frac{8}{33}$ in the two-qubit scenario) hold {\it constant} over the Bloch radii ($r$) of the single-qubit subsystems, jumping to 1 at the pure state boundaries ($r=1$). Here, firstly, we present evidence that in the qubit-qutrit case, the separability probability is uniformly distributed, as well, over the {\it generalized} Bloch radius ($R$) of the qutrit subsystem. While the qubit (standard) Bloch vector is positioned in three-dimensional space, the qutrit generalized Bloch vector lives in eight-dimensional space. The radii variables $r$ and $R$ themselves are the lengths/norms (being square roots of {\it quadratic} Casimir invariants) of these ("coherence") vectors. Additionally, we find that not only are the qubit-qutrit separability probabilities invariant over the quadratic Casimir invariant of the qutrit subsystem, but apparently also over the {\it cubic} one--and similarly the case, more generally, with the use of random induced measure. We also investigate two-qutrit ($3 \times 3$) and qubit-{\it qudit} ($2 \times 4$) systems--with seemingly analogous {\it positive-partial-transpose}-probability invariances holding over what have been termed by Altafini, the {\it partial} Casimir invariants of these systems.

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