Information geometry and its applications

Pescara - 1-5 July 2002


Abstract

July 4 - 16.40-17.30
A. Jencova - Information geometry in the standard representation of matrix spaces

The algebra of operators acting on a Hilbert space is standardly represented on the space W of Hilbert-Schmidt operators. The aim of the present contribution is to show how (in finite dimensions) the basic structures of quantum information geometry are lifted to W. It was shown by Dittmann and Uhlmann that the monotone Riemannian metrics are related to certain real vector subspaces in W. We show that there is a natural duality of such subspaces, which suggests a duality of the corresponding metrics. We also introduce dual parallel transports, related to the exponential and mixture connections. As examples, we treat the smallest (Bures) and the largest monotone metric and the smallest WYD metric. In these cases, we also show that the corresponding one-dimensional exponential families are related to positive cones in W. [an error occurred while processing this directive]