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Article overview
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8/33 and 8/17 -- Two-Qubit Complex and Real Hilbert-Schmidt Separability
Probabilities | | Paul B. Slater
; | | Date: |
27 Apr 2007 | | Subject: | Quantum Physics (quant-ph) | | Abstract: | We develop mutually supporting numerical and theoretical arguments for
modifying the two ansatze, involving incomplete beta functions, advanced in our
recent study, ``Two-qubit separability probabilities and beta functions’’
(quant-ph/0609006). That analysis was concerned with determining -- using a new
Bloore-parameterization-based methodology -- the Hilbert-Schmidt volumes of the
convex sets of separable 9-dimensional real and 15-dimensional complex
two-qubit states (representable by 4 x 4 density matrices r). We are led here
to adjust the two ansatze in the following manner. Firstly, the ``separability
function’’ in the real case is taken to be proportional to the incomplete beta
function B(v,1/2,2) where v= (r_{11} r_{44})/(r_{22} r_{33}). Then, in the
complex case -- conforming to a pattern exhibiting the Dyson indices (1, 2, 4)
of random matrix theory that we find in lower-dimensional exact analyses, we
conduct at considerable length -- the corresponding separability function is
taken to be simply proportional to B(v,1/2,2)^2. Exact requisite integrations
of products of the separability functions and associated jacobians over v in
[0,1] can now be readily performed. A data-fitting choice of scaling constants
-- coupled with the Hilbert-Schmidt (separable and nonseparable) volumes
computed by Zyzckowski and Sommers (quant-ph/0302197) -- then leads to the
probabilistic conclusions enunciated in the title. We also conduct further
analyses directed to analogous issues in higher-dimensional quantum systems
(qubit-qutrit,...). | | Source: | arXiv, arxiv.0704.3723 | | Services: | Forum | Review | PDF | Favorites | |
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