Narrow Results

We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0\le r_A,r_B\le 1$) 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 HS (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 [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A48 (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 Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form $p(r_A,\frac{1}{2})$. 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.

We conduct a pair of quasirandom estimations of the separability probabilities with respect to 10 measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parametrization. The measures include the (nonmonotone) 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 $1-\frac{256}{27{\pi }^2}\approx 0.0393251$. 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, $\frac{1}{9}(593-60{\pi }^2)\approx 0.0915262$, we now introduce — cannot be examined through the numerical approach adopted, at least perhaps not without some truncation procedure for extreme values.

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 two-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 $\alpha$ employed.