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A zero-mean entanglement index and related Hilbert-Schmidt moment computations for real two-qubit density matrices | | Paul B. Slater
; | | Date: |
27 Jul 2010 | | Abstract: | We study the moments of probability distributions generated by certain
determinantal functions of generic two-qubit density matrices (rho) with real
entries over the associated nine-dimensional convex domain, assigned
Hilbert-Schmidt measure. It is found that the mean of the (nonnegative)
determinant |rho| is 1/2288, the mean of the determinant of the partial
transpose |rho^{PT}|--negative values indicating entanglement--is -1/858, while
the mean of the product of these two determinants is zero. We ascertain the
exact values--also rational numbers--of the succeeding eight moments of
|rho^{PT}|. At intermediate steps in the derivation of the m-th moment,
rational functions C_{2 j}(m) emerge, yielding the coefficients of the 2j-th
power of even polynomials of total degree 4 m. These functions possess poles at
finite series of consecutive half-integers, and certain (trivial) roots at
finite series of consecutive natural numbers. The (nontrivial) dominant roots
of C_{2 j}(m) appear to converge to the same half-integer values, as j
increases. If formulas for C_{2j}(m) can be developed for arbitrary j--we have
them for j<9, then, the desired Hilbert-Schmidt separability probability would
be computable to high accuracy. We reproduce the (linearly transformed) first
nine moments of |rho^{PT}| quite closely by a certain (two-parameter) beta
distribution, and more so by a three-parameter (Libby-Novick) extension of it.
The first two moments of |rho^{PT}|--when employed in the one-sided Chebyshev
inequality--give an upper bound of 30397/34749 = 0.874759 on the
Hilbert-Schmidt separability probability of real two-qubit density matrices. We
ascertain by numerical methods that the Hilbert-Schmdit zero-mean property of
|rho| |rho^{PT}|=|rho rho^{PT}| does not extend to the Bures (minimal monotone)
counterpart. | | Source: | arXiv, 1007.4805 | | Services: | Forum | Review | PDF | Favorites | |
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