Coend Optics for Quantum Combs

TL;DR

This paper compares two categorical definitions of quantum combs, showing their equivalence under certain conditions and extending the framework to n-combs, with implications for quantum theory modeling.

Contribution

It establishes a formal equivalence between intensional and extensional definitions of quantum combs using category theory and extends the framework to polycategories of n-combs.

Findings

Full and bijective functor relates the two definitions.

Conditions identified for categorical isomorphism.

Framework extended to polycategories of n-combs.

Abstract

We compare two possible ways of defining a category of 1-combs, the first intensionally as coend optics and the second extensionally as a quotient by the operational behaviour of 1-combs on lower-order maps. We show that there is a full and bijective on objects functor quotienting the intensional definition to the extensional one and give some sufficient conditions for this functor to be an isomorphism of categories. We also show how the constructions for 1-combs can be extended to produce polycategories of n-combs with similar results about when these polycategories are equivalent. The extensional definition is of particular interest in the study of quantum combs and we hope this work might produce further interest in the usage of optics for modelling these structures in quantum theory.