Exact Bures probabilities that two quantum bits are classically correlated
Abstract
In previous studies, we have explored the ansatz that the volume elements of the Bures metrics over quantum systems might serve as prior distributions, in analogy with the (classical) Bayesian role of the volume elements (``Jeffreys' priors'') of Fisher information metrics. Continuing this work, we obtain exact Bures prior probabilities that the members of certain lowdimensional subsets of the fifteendimensional convex set of density matrices are separable or classically correlated. The main analytical tools employed are symbolic integration and a formula of Dittmann (J. Phys. A 32, 2663 (1999)) for Bures metric tensors. This study complements an earlier one (J. Phys. A 32, 5261 (1999)) in which numerical (randomization)  but not integration  methods were used to estimate Bures separability probabilities for unrestricted and density matrices. The exact values adduced here for pairs of quantum bits (qubits), typically, well exceed the estimate ( ) there, but this disparity may be attributable to our focus on special lowdimensional subsets. Quite remarkably, for the q= 1 and states inferred using the principle of maximum nonadditive (Tsallis) entropy, the Bures probabilities of separability are both equal to . For the Werner qubitqutrit and qutritqutrit states, the probabilities are vanishingly small, while in the qubitqubit case it is .
 Publication:

European Physical Journal B
 Pub Date:
 October 2000
 DOI:
 10.1007/s100510070126
 arXiv:
 arXiv:quantph/9911058
 Bibcode:
 2000EPJB...17..471S
 Keywords:

 03.67.a;
 03.65.Bz;
 02.40.Ky;
 02.50.r;
 Quantum information;
 Riemannian geometries;
 Probability theory stochastic processes and statistics;
 Quantum Physics;
 Mathematical Physics;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 Seventeen pages, LaTeX, eleven postscript figures. In this version, subsequent (!) to publication in European Physical Journal B, we correct the (1,1)entries of the 4 x 4 matrices given in formulas (6) and (7), that is, the numerators should both read v^2  x^2  y^2  z^2, rather than v^2  x^2 + y^2 + z^2