We extend to arbitrarily coupled pairs of qubits (two-state quantum systems) and qutrits (three-state quantum systems) our earlier study (Slater 2002 Quantum Inf. Process. 1 387), which was concerned with the simplest instance of entangled quantum systems, pairs of qubits. As in that analysis—again on the basis of numerical (quasi-Monte Carlo) integration results, but now in a still higher-dimensional space (35D versus 15D)—we examine a conjecture that the Bures/SD (statistical distinguishability) probability that arbitrarily paired qubits and qutrits are separable (unentangled) has a simple exact value, , where u = 2 20 × 3 3 × 5 × 7 and v = 19 × 23 × 29 × 31 × 37 × 41 × 43 (the product of consecutive primes). This is considerably less than the conjectured value of the Bures/SD probability, , in the qubit–qubit case. Both of these conjectures, in turn, rely upon ones to the effect that the SD volumes of separable states assume certain remarkable forms, involving 'primorial' numbers. We also estimate the SD area of the boundary of separable qubit–qutrit states, and provide preliminary calculations of the Bures/SD probability of separability in the general qubit–qubit–qubit and qutrit–qutrit cases.
We extend to arbitrarily coupled pairs of qubits (two-state quantum systems) and qutrits (three-state quantum systems) our earlier study (Slater 2002 Quantum Inf. Process. 1 387), which was concerned with the simplest instance of entangled quantum systems, pairs of qubits. As in that analysis—again on the basis of numerical (quasi-Monte Carlo) integration results, but now in a still higher-dimensional space (35D versus 15D)—we examine a conjecture that the Bures/SD (statistical distinguishability) probability that arbitrarily paired qubits and qutrits are separable (unentangled) has a simple exact value, , where u = 2 20 × 3 3 × 5 × 7 and v = 19 × 23 × 29 × 31 × 37 × 41 × 43 (the product of consecutive primes). This is considerably less than the conjectured value of the Bures/SD probability, , in the qubit–qubit case. Both of these conjectures, in turn, rely upon ones to the effect that the SD volumes of separable states assume certain remarkable forms, involving 'primorial' numbers. We also estimate the SD area of the boundary of separable qubit–qutrit states, and provide preliminary calculations of the Bures/SD probability of separability in the general qubit–qubit–qubit and qutrit–qutrit cases.
In a previous study (quant-ph/0207181), we formulated a conjecture that arbitrarily coupled qubits (describable by 4 x 4 density matrices) are separable with an a priori probability of 8/(11 π^2) = 0.0736881. For this purpose, we employed the normalized volume element of the Bures (minimal monotone) metric as a probability distribution over the fifteen-dimensional convex set of 4 x 4 density matrices. Here, we provide further/independent (quasi-Monte Carlo numerical integration) evidence of a stronger nature (giving an estimate of 0.0736858 vs. 0.0737012 previously) for this conjecture. Additionally, employing a certain ansatz, we estimate the probabilities of separability based on certain other monotone metrics of interest. However, we find ourselves, at this point, unable to convincingly conjecture exact simple formulas for these new (smaller) probabilities.
In a number of previous studies, we have investigated the use of the volume element of the Bures (minimal monotone) metric -- identically, one-fourth of the statistical distinguishability (SD) metric -- as a natural measure over the (n^2-1)-dimensional convex set of n x n density matrices. This has led us for the cases n = 4 and 6 to estimates of the prior (Bures/SD) probabilities that qubit-qubit and qubit-qutrit pairs are separable. Here, we extend this work from such bipartite systems to the tripartite "laboratory'' quantum systems possessing U x U x U symmetry recently constructed by Eggeling and Werner (Phys. Rev. A 63 [2001], 042324). We derive the associated SD metric tensors for the three-qubit and three-qutrit cases, and then obtain estimates of the various related Bures/SD probabilities using Monte Carlo methods.
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
We detect a certain pattern of behavior of separability probabilities [Formula: see text] for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where [Formula: see text] and [Formula: see text] are the Bloch radii ([Formula: see text]) of the qubit reduced states ([Formula: see text]). We observe a relative repulsion of radii effect, that is [Formula: see text], except for rather narrow “crossover” intervals [Formula: see text]. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional [Formula: see text]-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in [Formula: see text]-dimensions, for [Formula: see text], with [Formula: see text] corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) [Formula: see text], the [Formula: see text]-states [Formula: see text], and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two [Formula: see text]-states cases, we derive analytical results; for [Formula: see text], 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 [Formula: see text]) as [Formula: see text] increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A 48 (2015) 035306.] from a univariate ([Formula: see text]) framework — in which they found separability probabilities to hold constant with [Formula: see text] — to a bivariate ([Formula: see text]) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form [Formula: see text]. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.
Due to recent important work of \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski and Sommers [J. Phys. A 36, 10115 (2003); 36, 10083 (2003)], exact formulas are available (in terms of both the Hilbert-Schmidt and the Bures metrics) for the $({n}^{2}\ensuremath{-}1)$-dimensional and $[n(n\ensuremath{-}1)∕2\ensuremath{-}1]$-dimensional volumes of the complex and real $n\ifmmode\times\else\texttimes\fi{}n$ density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, two-qubit systems. Making use of the density matrix $(\ensuremath{\rho})$ parametrization of Bloore [J. Phys. A 9, 2059 (1976)], we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, $\ensuremath{\nu}={\ensuremath{\rho}}_{11}{\ensuremath{\rho}}_{44}∕{\ensuremath{\rho}}_{22}{\ensuremath{\rho}}_{33}$. The associated integrand in each case is the product of a known Jacobian (highly oscillatory near $\ensuremath{\nu}=1$) and a certain unknown univariate function, which our extensive numerical (quasi Monte Carlo) computations indicate is very closely proportional to an (incomplete) Beta function ${B}_{\ensuremath{\nu}}(a,b)$, with a=1∕2, b=3 in the real case, and a=26∕5,b=3∕2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.
Paralleling our recent computationally intensive (quasi-Monte Carlo) work for the case N=4 (e-print quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski (e-print quant-ph/0304041) for the $({N}^{2}\ensuremath{-}1)$-dimensional volume and $({N}^{2}\ensuremath{-}2)$-dimensional hyperarea of the (separable and nonseparable) $N\ifmmode\times\else\texttimes\fi{}N$ density matrices, based on the Bures (minimal monotone) metric---and also their analogous formulas (e-print quant-ph/0302197) for the (nonmonotone) flat Hilbert-Schmidt metric. With the same seven 109 well-distributed (``low-discrepancy'') sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically rank-6 (rank-5) density matrices. The (rank-6) separability probabilities obtained based on the 35-dimensional volumes appear to be---independently of the metric (each of the seven inducing Haar measure) employed---twice as large as those (rank-5 ones) based on the 34-dimensional hyperareas. (An additional estimate---33.9982---of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit simple exact formulas to our estimates of the Hilbert-Schmidt separable volumes and hyperareas in both the N=4 and N=6 cases.