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28 March 2024
 
  » arxiv » 1605.6459

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Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems
Paul B. Slater ;
Date 20 May 2016
AbstractWe 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 $[ ilde{r}_A,frac{1}{2}]$. 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 $4 imes K$-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 $ ilde{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.
Source arXiv, 1605.6459
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