Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity

Abstract

Quantum theory satisfies the mathematically powerful property of self-duality, meaning that its state and effect spaces are made isomorphic by an inner product. This is why both quantum states and effects can be represented by positive Hermitian operators. The seminal Koecher-Vinberg theorem shows that self-duality is a rather special property: any system that is self-dual and also homogeneous, meaning that the group of symmetries of its cone of unnormalised states acts transitively on the strictly positive states, must be isomorphic to a Euclidean Jordan algebra. EJAs have been classified and are known to be just a slight generalisation of quantum systems. This is why the Koecher-Vinberg theorem, and hence self-duality and homogeneity, have been used in several reconstructions of quantum theory from first principles. While homogeneity has an operational derivation from the ability to steer states, self-duality currently lacks such a clear operational motivation. In this paper we prove an alternative to the Koecher-Vinberg theorem, substituting for self-duality the more operational property of pure transitivity: symmetries of the normalised state space act transitively on the pure states. We show that any system that is homogeneous and satisfies pure transitivity must be self-dual, and hence isomorphic to an EJA. Together with various ways of singling out the complex matrix algebras from among EJAs, and known ways of ruling out classicality in these, this yields several new and concise reconstructions of quantum theory. For example, quantum systems, classical systems, and composites of these are the only ones that are homogeneous, satisfy pure transitivity, and allow locally tomographic composites; fully quantum systems are the only ones that are homogeneous, have continuous pure transitivity, and allow a correspondence between observables and generators of reversible transformations.