Uniqueness of Composition in Quantum Theory and Linguistics

Bob Coecke
(University of Oxford)
Fabrizio Genovese
(University of Oxford)
Stefano Gogioso
(University of Oxford)
Dan Marsden
(University of Oxford)
Robin Piedeleu
(University of Oxford)

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 semirings—as modelled by categories of free finite-dimensional modules—and we prove that the only bilinear compact-closed symmetric monoidal structure is the canonical one (up to linear monoidal equivalence). Our results apply to conventional quantum theory and other toy theories of interest in the literature, such as real quantum 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 histograms—admit an (essentially) unique compatible pregroup grammar.

In Bob Coecke and Aleks Kissinger: Proceedings 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July 2017, Electronic Proceedings in Theoretical Computer Science 266, pp. 249–257.
Authors of this work are listed in alphabetical order.
Published: 27th February 2018.

ArXived at: http://dx.doi.org/10.4204/EPTCS.266.17 bibtex PDF
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