Entropy distance: New quantum phenomena

Abstract

We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.