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Two-Qubit Hilbert-Schmidt Separability Functions and Probabilities for Full-Dimensional Even-Dyson-Index Scenario | | Paul B. Slater
; | | Date: |
18 Oct 2007 | | Abstract: | We extend the findings and analyses of our two recent studies (Phys. Rev. A,
75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates
of the separability function based on the (Euclidean, flat) Hilbert-Schmidt
(HS) metric for the 27-dimensional convex set of quaternionic two-qubit
systems. The estimated function appears to be strongly consistent with our
previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the
quaternionic (beta=4) separability function should be exactly proportional to
the square of the separability function for the 15-dimensional convex set of
two-qubit complex (beta=2) systems, as well as the fourth power of the
separability function for the 9-dimensional convex set of two-qubit real
(beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2
((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22}
rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus,
supplement our previous assertion that the HS separability probability of the
two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic
counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze
the odd beta=1 and 3 cases. | | Source: | arXiv, 0710.3549 | | Services: | Forum | Review | PDF | Favorites | |
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