Compactly accessible categories and quantum key distribution

TL;DR

This paper introduces compactly accessible categories, extending compact categories to infinite dimensions using factorisation systems, and models quantum key distribution protocols categorically.

Contribution

It defines compactly accessible categories, enabling infinite-dimensional modeling while preserving key properties, and applies this framework to quantum key distribution.

Findings

Successfully models quantum key distribution categorically.

Extends the duals functor from compact to compactly accessible categories.

Provides a categorical proof of protocol correctness.

Abstract

Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finite-dimensional, they cannot accomodate (co)limit-based constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choice-of-duals functor on the compact part extends canonically to the whole compactly accessible category. As an example, we model a quantum key distribution protocol and prove its correctness categorically.