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Radial and Azimuthal Profiles of Two-Qubit/Rebit Hilbert-Schmidt Separability Probabilities | | Paul B. Slater
; | | Date: |
25 Oct 2010 | | Abstract: | We reduce the long-standing problem of ascertaining the Hilbert-Schmidt
probability that a generic pair of qubits is separable to that of determining
the specific nature of a one-dimensional (separability) function of the radial
coordinate (r) of the unit ball in 15-dimensional Euclidean space, and
similarly for a generic pair of rebits, using the 9-dimensional unit ball.
Separability probabilities, could, then, be directly obtained by integrating
the products of these functions (which we numerically estimate) with jacobian
factors of r^m over r in [0,1], with m=17 for the two-rebit case, and m=29 in
the two-qubit instance. We, further, repeat the analyses, but for the
replacement of r as the free variable, by the azimuthal angle phi in [0,2
pi]-with the associated jacobian factors now being, trivially, unity. So, the
separability probability estimates, then, become simply the areas under the
associated separability function curves. These curves appear to consist of flat
baselines plus (irrelevant for the separability probability question)
sinusoidal oscillations about the baseline. For our analyses, we employ an
interesting Cholesky-decomposition parameterization of the 4 x 4 density
matrices. In earlier, similarly-motivated studies of ours, we have relied upon
different parameterizations-in particular, those based on Euler angles and on
Bloore/correlations--to obtain analogous (univariate) separability functions.
All these several sets of results-although quite intriguing in their highly
particular, interesting natures-are not yet definitive in fully resolving the
two-qubit/rebit separability probability problem. | | Source: | arXiv, 1010.5180 | | Services: | Forum | Review | PDF | Favorites | |
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