Quantum Lunch: Quantum information on Jordan algebras

Speaker: Peter Harremoës

Title: Quantum information on Jordan algebras

Abstract:

Quantum relative entropy play a central role in quantum information theory, and we will characterize quantum relative entropy as a Bregman divergence that satisfies five equivalent conditions. The aim is to generalize this result from density matrices on complex Hilbert spaces to more general settings. We give give a formal definition of an abstract state space as a convex set. We demonstrate that even on general state spaces any state can be written as a convex combination of orthogonal states. A decomposition of a state as a convex combination of orthogonal states need not be unique, but for some convex sets the coefficients in the convex combination are unique. Convex sets with this property will be called spectral sets and they are the abstract state spaces where information theoretic quantities like quantum relative entropy can be defined. Apparently all spectral sets can be embedded in Jordan algebras and in this sense Jordan algebras appear to be a natural environment for doing abstract quantum information theory. The talk will contain a short introduction to Jordan algebras. Some preliminary results related to this talk can be found in arXiv:1607.02259 .