
Journal of Convex Analysis 21 (2014), No. 2, 339399 Copyright Heldermann Verlag 2014 Information Topologies on NonCommutative State Spaces Stephan Weis Max Planck Institute for Mathematics, Inselstraße 22, 04103 Leipzig, Germany sweis@mis.mpg.de We define an information topology (Itopology) and a reverse information topology (rItopology) on the state space of a C*subalgebra of Mat(n,C) in terms of sequential convergence with respect to the relative entropy. Open disks with respect to the relative entropy define a base for the topology. This was not evident since Csiszár has shown in the 1960's that the analogue is wrong for probability measures on a countably infinite set. The Itopology is strictly finer than the norm topology, it disconnects the convex state space into its faces. The rItopology is intermediate and it allows to complete two fundamental theorems of information geometry to the full state space, by taking the closure in the rItopology. The norm topology can be too coarse for this aim but for commutative algebras it equals the rItopology, so the difference belongs to the domain of quantum theory. We apply our results to the maximization of the von Neumann entropy under linear constraints and to the maximization of quantum correlations. Keywords: Relative entropy, information topology, exponential family, convex support, Pythagorean theorem, projection theorem, maximum entropy, mutual information. MSC: 81P45, 81P16, 54D55, 94A17, 90C26 [ Fulltextpdf (1129 KB)] for subscribers only. 