Symmetry and Composition in Probabilistic Theories

The past decade has seen a remarkable resurgence of the old programme of finding more or less a priori axioms for the mathematical framework of quantum mechanics. The new impetus comes largely from quantum information theory; in contrast to work in the older tradition, which tended to concentrate on structural features of individual quantum systems, the newer work is marked by an emphasis on systems in interaction. Within this newer work, one can discern two distinct approaches: one is “top-down”, and attempts to capture in category-theoretic terms what is distinctive about quantum information processing. The other is “bottom up”, attempting to construct non-classical models and theories by hand, as it were, and then characterizing those features that mark out quantum-like behavior. This paper blends these approaches. We present a constructive, bottom-up recipe for building probabilistic theories having strong symmetry properties, using as data any uniform enlargement of the symmetric group S(E) of any set, to a larger group G(E). Subject to some natural conditions, our construction leads to a monoidal category of fully symmetric test spaces, in which the monoidal product is “non-signaling”.