Random matrix techniques in quantum information theory

The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels

Exact Bures Probabilities that Two Quantum Bits are Classically Correlated

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 to the (classical) Bayesian role of the volume elements ("Jeffreys' priors") of Fisher information metrics. Continuing this work, we obtain exact Bures probabilities that the members of certain low-dimensional subsets of the fifteen-dimensional convex set of 4 x 4 density matrices are separable or classically correlated. The main analytical tools employed are symbolic integration and a formula of Dittmann (quant-ph/9908044) for Bures metric tensors. This study complements an earlier one (quant-ph/9810026) in which numerical (randomization) --- but not integration --- methods were used to estimate Bures separability probabilities for unrestricted 4 x 4 or 6 x 6 density matrices. The exact values adduced here for pairs of quantum bits (qubits), typically, well exceed the estimate (.1) there, but this disparity may be attributable to our focus on special low-dimensional subsets. Quite remarkably, for the q = 1 and q = 1/2 states inferred using the principle of maximum nonadditive (Tsallis) entropy, the separability probabilities are both equal to 2^{1/2} - 1. For the Werner qubit-qutrit and qutrit-qutrit states, the probabilities are vanishingly small, while in the qubit-qubit case it is 1/4.Comment: 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^