  The notion of Bregman divergence and sufficiency will be defined on general
convex state spaces. It is demonstrated that only spectral sets can have a
Bregman divergence that satisfies a sufficiency condition. Positive elements
with trace 1 in a Jordan algebra are examples of spectral sets, and the most
important example is the set of density matrices with complex entries. It is
conjectured that information theoretic considerations lead directly to the
notion of Jordan algebra under some regularity conditions.
