# Universal Properties in Quantum Theory

## TL;DR

This paper explores the universal categorical properties of quantum channels, specifically CPTP maps, showing they are the affine reflection of isometries within a monoidal category framework.

## Contribution

It formalizes the universal property of quantum channels in category theory, connecting CPTP maps to isometries through affine reflection.

## Key findings

- CPTP maps form a universal monoidal category with a terminal unit.
- Quantum channels are derived from isometries via affine reflection.
- Provides a categorical foundation for understanding quantum channels.

## Abstract

We argue that notions in quantum theory should have universal properties in the sense of category theory. We consider the completely positive trace preserving (CPTP) maps, the basic notion of quantum channel. Physically, quantum channels are derived from pure quantum theory by allowing discarding. We phrase this in category theoretic terms by showing that the category of CPTP maps is the universal monoidal category with a terminal unit that has a functor from the category of isometries. In other words, the CPTP maps are the affine reflection of the isometries.