Entropy distance: new quantum phenomena

The relative entropy distance of a state from an exponential family is important in information theory and statistics. Two-dimensional examples in the algebra of complex $3 \times 3$-matrices reveal that the mean value set of an exponential family has typically non-exposed faces. The Staffelberg family stands out of these examples due to a discontinuous entropy distance.

We meet these phenomena e.g. in optimization problems on the state space (including singular states). They do not occur in the probabilistic case of an abelian $^{\ast}$-subalgebra of complex $N \times N$-matrices. Analogues of probability theory exist in the non-abelian quantum case, though: The entropy distance from an exponential family in a finite-dimensional ${C^{\ast}}$-algebra is given by a projection to an extension of the family. An optimal form of the Pythagorean theorem of relative entropy holds for this extension. We conclude with applications.

Part of this work is jointly with Andreas Knauf.