2010/03/19 by Paul B. Slater, Slater, Paul B.
Chemistry · Engineering · Mathematics · Physics and Astronomy · #15B48 #60E05 #81P40 #81P45 #Advanced Mathematical Theories and Applications #FOS: Physical sciences #Mathematical Inequalities and Applications #Molecular spectroscopy and chirality #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Random Matrices and Applications #graph theory and CDMA systems #msc:15B48 #msc:60E05 #msc:81P40 #msc:81P45 #quant-ph
paper · pdf · doi:10.48550/arxiv.1003.3839
15 pages, 4 figures
arxiv created 2010/03/19 · openalex publication_date 2010/03/19 · arxiv updated 2010/03/22 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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 C2 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 C2 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.