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Article overview
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Two-Qubit Separability Probabilities: A Concise Formula | | Paul B. Slater
; | | Date: |
7 Sep 2012 | | Abstract: | We report a concise answer--in the case of 2 x 2 systems--to the fundamental
quantum-information-theoretic question as to "the volume of separable states"
posed by Zyczkowski, Horodecki, Sanpera and Lewenstein (Phys. Rev. A, 58, 883
[1998]). We proceed by applying the Mathematica command FindSequenceFunction to
a series of conjectured Hilbert-Schmidt generic 2 x 2 (rational-valued)
separability probabilities p(a), a = 1, 2,...,32, with a = 1 indexing standard
two-qubit systems, and a = 2, two-quater(nionic)bit systems. These 32 inputted
values of p(a)--as well as 32 companion non-inputted values for the
half-integers, a = 1/2 (two-re[al]bit) systems), 3/2,..., 63/2, are advanced on
the basis of high-precision probability-distribution-reconstruction
computations, employing 7,501 determinantal moments of partially transposed 4 x
4 density matrices. The function P(a) given by application of the command fully
reproduces both of these 32-length sequences, and an equivalent outcome is
obtained if the half-integral series is the one inputted. The lengthy
expression (containing six hypergeometric functions) obtained for P(a) is,
then, impressively condensed (by Qing-Hu Hou and colleagues), using
Zeilberger’s algorithm. For generic (9-dimensional) two-rebit systems, P(1/2) =
29/64, (15-dimensional) two-qubit systems, P(1) = 8/33, (27-dimensional)
two-quaterbit systems, P(2) = 26/323, while for generic classical
(3-dimensional) systems, P(0)=1. | | Source: | arXiv, 1209.1613 | | Services: | Forum | Review | PDF | Favorites | |
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