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.
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