Physics – Mathematical Physics
Scientific paper
2010-03-29
Physics
Mathematical Physics
53 pages, 4 figures
Scientific paper
We investigate the I-topology and the rI-topology on the state spaces of a C*-subalgebra of Mat(n,C). These are defined in terms of convergence of the relative entropy. The I-topology includes the rI-topology, which includes the norm topology. These topologies share some properties with a metric topology, in particular their open sets are unions of disks arising from the relative entropy. On the other hand, they are quite distinct from the norm topology, e.g. they recognize the convex geometry of the state space and they are not second countable unless the algebra is commutative. In quantum information geometry there are a well-known Pythagorean theorem and projection theorem, valid for exponential families of invertible states. We complete these theorems to the rI-closure of an exponential family, which contains states of diverse support and which can be strictly included into the norm closure. We discuss the non-commutative feature of a discontinuous entropy distance (the infimum of the relative entropy) from an exponential family. Applications in quantum information theory include the first complete solution of the von Neumann entropy maximization under linear constraints and two nec- essary conditions for local maximizers of the entropy distance.
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