Information topologies on non-commutative state spaces

Abstract

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). These topologies arise from sequential convergence with respect to the relative entropy. We prove that open disks, with respect to the relative entropy, define a base for them, while Csiszar has shown in 1967 that the analogue is wrong for probability measures on a countably infinite set. The I-topology is finer than the norm topology, it disconnects the convex state space into its faces. The rI-topology is intermediate between these topologies. We complete two fundamental theorems of information geometry to the full state space, by taking the closure in the rI-topology. The norm topology is too coarse for this aim only for a non-commutative algebra, so its discrepancy to the rI-topology belongs to the quantum domain. We apply our results to the maximization of the von Neumann entropy under linear constraints and to the maximization of quantum correlations.