# Quantum channels as a categorical completion

## TL;DR

This paper establishes a categorical framework connecting pure and mixed quantum states using distributive monoidal categories, showing that quantum channels form a canonical completion of pure quantum operations, and extends this to topological settings.

## Contribution

It introduces a categorical foundation for quantum channels as a completion of pure operations and generalizes it to topological contexts.

## Key findings

- Quantum channels form a canonical completion of pure quantum operations.
- The category of completely positive trace-preserving maps is a completion of vector spaces and isometries.
- The operator norm topology on quantum channels is induced by the norm topology on isometries.

## Abstract

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.