Comparative Noninformativities of Quantum Priors Based on Monotone Metrics

We consider a family of prior probability distributions of particular interest, all being defined on the three-dimensional convex set of two-level quantum systems. Each distribution is, following recent work of Petz and Sudar, taken to be proportional to the volume element of a monotone metric on that Riemannian manifold. We apply an entropy-based test (a variant of one recently developed by Clarke) to determine which of two priors is more noninformative in nature. This involves converting them to posterior probability distributions based on some set of hypothesized outcomes of measurements of the quantum system in question. It is, then, ascertained whether or not the original relative entropy (Kullback-Leibler distance) between a pair of priors increases or decreases when one of them is exchanged with its corresponding posterior. The findings lead us to assert that the maximal monotone metric yields the most noninformative (prior) distribution and the minimal monotone (that is, the Bures) metric, the least. Our conclusions both agree and disagree, in certain respects, with ones recently reached by Hall, who relied upon a less specific test criterion than our entropy-based one.Comment: 7 pages, LaTeX, minor changes, to appear in Physics Letters

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 $p(r_A,\frac{1}{2})$.Comment: 24 pages, 17 figures--revised--to appear in International Journal of Quantum Informatio