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Formulas for Generalized Two-Qubit Separability Probabilities
To begin, we find certain formulas , for . These yield that part of the total
separability probability, , for generalized (real, complex,
quaternionic,\ldots) two-qubit states endowed with random induced measure, for
which the determinantal inequality holds. Here
denotes a density matrix, obtained by tracing over the pure states
in -dimensions, and , its partial transpose.
Further, is a Dyson-index-like parameter with for the
standard (15-dimensional) convex set of (complex) two-qubit states. For ,
we obtain the previously reported Hilbert-Schmidt formulas, with (the real
case) , (the standard complex case)
, and (the quaternionic case) ---the three simply equalling . The factors
are sums of polynomial-weighted generalized hypergeometric
functions , , all with argument . We find number-theoretic-based formulas for the upper
() and lower () parameter sets of these functions and, then,
equivalently express in terms of first-order difference
equations. Applications of Zeilberger's algorithm yield "concise" forms,
parallel to the one obtained previously for . For
nonnegative half-integer and integer values of , has
descending roots starting at . Then, we (C. Dunkl and I) construct
a remarkably compact (hypergeometric) form for itself. The
possibility of an analogous "master" formula for is, then,
investigated, and a number of interesting results found.Comment: 78 pages, 5 figures, 15 appendices, to appear in Adv. Math.
Phys--verification in arXiv:1701.01973 of 8/33-two-qubit Hilbert-Schmidt
separability probability conjecture note
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