Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes

Details Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes

2010-03-19

arxiv.org/abs/1003.3839v1

Physics

Quantum Physics

15 pages, 4 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-354811