Classical Structures Based on Unitaries

TL;DR

This paper explores the relationship between classical structures in categorical quantum mechanics and self-similarity in logic and computation, revealing conditions under which they coincide and providing examples and coherence results.

Contribution

It demonstrates that classical structures and self-similarity can be connected through canonical isomorphisms, with new coherence results and matrix representation analogues.

Findings

Distinct classical structures are linked to self-similarity via canonical isomorphisms.

Examples show how self-similar structures determine matrix representations.

Analogues of linear algebra concepts like basis change and diagonalisation are developed.

Abstract

Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide. Provided all definitions are strict in the categorical sense, we show that this can never be the case. However, allowing for the defining axioms to be taken up to canonical isomorphism, a close connection between the classical structures of categorical quantum mechanics, and the categorical property of self-similarity familiar from logical and computational models becomes apparent. The required canonical isomorphisms are non-trivial, and mix both typed (multi-object) and untyped (single-object) tensors and structural isomorphisms; we give coherence results that justify this approach. We then give a class of examples where distinct self-similar structures at an object determine distinct matrix representations of arrows, in the same way as classical structures determine matrix representations in Hilbert space. We also give analogues of familiar notions from linear algebra in this setting such as changes of basis, and diagonalisation.