For convex optimization problems Bregman divergences appear as regret
functions. Such regret functions can be defined on any convex set but if a
sufficiency condition is added the regret function must be proportional to
information divergence and the convex set must be spectral. Spectral set are
sets where different orthogonal decompositions of a state into pure states have
unique mixing coefficients. Only on such spectral sets it is possible to define
well behaved information theoretic quantities like entropy and divergence. It
is only possible to perform measurements in a reversible way if the state space
is spectral. The most important spectral sets can be represented as positive
elements of Jordan algebras with trace 1. This means that Jordan algebras
provide a natural framework for studying quantum information. We compare
information theory on Hilbert spaces with information theory in more general
Jordan algebras, and conclude that much of the formalism is unchanged but also
identify some important differences.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1701.0101