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

TL;DR

This paper investigates how the probability of two-qubit states being separable depends on the Bloch radii of their subsystems, revealing patterns and crossover behaviors across various models and measures.

Contribution

It extends previous univariate analyses to a bivariate framework, providing analytical and numerical insights into separability probabilities as functions of Bloch radii.

Findings

Identifies a pattern of relative radii repulsion in separability probabilities.

Derives analytical formulas for certain X-states models.

Observes crossover regions expanding with increasing K.

Abstract

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 $p(r_A,\frac{1}{2})$.