Universal quantum uncertainty relations between nonergodicity and loss of information

Quantum mechanics and information theory are among the most important scientific discoveries of the last century. Although these two areas initially developed separately it has emerged that they are in fact intimately related. In this review I will show how quantum information theory extends traditional information theory by exploring the limits imposed by quantum, rather than classical mechanics on information storage and transmission. The derivation of many key results uniquely differentiates this review from the "usual" presentation in that they are shown to follow logically from one crucial property of relative entropy. Within the review optimal bounds on the speed-up that quantum computers can achieve over their classical counter-parts are outlined using information theoretic arguments. In addition important implications of quantum information theory to thermodynamics and quantum measurement are intermittently discussed. A number of simple examples and derivations including quantum super-dense coding, quantum teleportation, Deutsch's and Grover's algorithms are also included.

With growing success in experimental implementations it is critical to identify a "gold standard" for quantum information processing, a single measure of distance that can be used to compare and contrast different experiments. We enumerate a set of criteria such a distance measure must satisfy to be both experimentally and theoretically meaningful. We then assess a wide range of possible measures against these criteria, before making a recommendation as to the best measures to use in characterizing quantum information processing.