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Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes | | Paul B. Slater
; | | Date: |
19 Mar 2010 | | Abstract: | The nonnegativity of the determinant of the partial transpose of a two-qubit
(4 x 4) density matrix is both a necessary and sufficient condition for its
separability. While the determinant is restricted to the interval [0,1/256],
the determinant of the partial transpose can range over [-1/16,1/256], with
negative values corresponding to entangled states. We report here the exact
values of the first nine moments of the probability distribution of the partial
transpose over this interval, with respect to the Hilbert-Schmidt (metric
volume element) measure on the nine-dimensional convex set of real two-qubit
density matrices. Rational functions C_{2 j}(m), yielding the coefficients of
the 2j-th power of even polynomials occurring at intermediate steps in our
derivation of the m-th moment, emerge. These functions possess poles at finite
series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain
(trivial) roots at finite series of consecutive natural numbers (m=0, 1,...).
Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same
half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first
two moments (mean and variance) found--when employed in the one-sided Chebyshev
inequality--give an upper bound of 30397/34749 = 0.874759 on the separability
probability of real two-qubit density matrices. We are able to report general
formulas for the m-th moment of the Hilbert-Schmidt probability distribution of
the density matrix determinant over [0,1/256], in the real, complex and
quaternionic two-qubit cases. | | Source: | arXiv, 1003.3839 | | Services: | Forum | Review | PDF | Favorites | |
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