Quantum correlations and distinguishability of quantum states

Abstract

A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.Comment: Review article, 103 pages, to appear in J. Math. Phys. 55 (special issue: non-equilibrium statistical mechanics, 2014

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

“…The quantum Chernoff metric also enjoys increasing popularity in the information approach in physics [39][40][41]. It appears under several different names, among which are (up to a factor of one forth) the Hellinger metric [4] or the Wigner-Yanase metric [21][22][23].…”

“…Further, the set of density matrices ρ should be endowed with a metric structure and thus explored as a Riemannian manifold. The realization of the set of density matrices ρ as a Riemannian manifold implies to define a relevant distance between two mixed quantum states [1,4]. The notion of distance has found applications in different fields.…”

“…The consequences of this fact are subject of considerable interest, see, e.g. [1][2][3][4] along with a number of references therein. The relation between operator monotone functions and monotone metrics allows to explorer in deep the relations between the different metrics in the context of quantum statistical mechanics and condensed matter physics.…”

“…In recent decade different concepts emerging from quantum-information geometry [1][2][3][4] have been intensively incorporated in condensed matter physics [5][6][7][8][9][10][11][12][13][14][15][16][17]. The underlying idea may be briefly reviewed as follows.…”

“…[6,10,[36][37][38] and references therein. The Bures metric d 2 B appears under different names: quantum Fisher information (apart from a numerical factor), SLD metric, fidelity susceptibility, etc [4,6,18,19].…”