Uniqueness of composition in quantum theory and linguistics

Abstract

We derive a uniqueness result for non-Cartesian composition of systems in a large class of process theories, with important implications for quantum theory and linguistics. Specifically, we consider theories of wavefunctions valued in commutative involutive semiringsas modelled by categories of free finite-dimensional semimodulesand we prove that the only bilinear compact-closed sym- metric monoidal structure is the canonical one (up to monoidal equivalence). Our results apply to conventional quantum theory and other toy theories of interest in the literature, such as real quan- tum theory, relational quantum theory, hyperbolic quantum theory and modal quantum theory. In computational linguistics they imply that linear models for categorical compositional distributional semantics (DisCoCat)such as vector spaces, sets and relations, and sets and histogramsadmit an (essentially) unique compatible pregroup grammar.