Information topologies on non-commutative state spaces
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Submission date: 24. Nov. 2011 (revised version: April 2013)
published in: Journal of convex analysis, 21 (2014) 2, p. 339-399
MSC-Numbers: 81P45, 81P16, 54D55, 94A17, 90C26
Keywords and phrases: non-commutative algebra, relative entropy, information topology, exponential family, convex support, Pythagorean theorem, projection theorem, von Neumann entropy
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We define an information topology (I-topology) and a reverse information topology (rI-topology) 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 I-topology is strictly finer than the norm topology, it disconnects the convex state space into its faces. The rI-topology is intermediate and it allows to complete two fundamental theorems of information geometry to the full state space, by taking the closure in the rI-topology. The norm topology can be too coarse for this aim but for commutative algebras it equals the rI-topology, 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.