Bennett and Stinespring, Together at Last

TL;DR

This paper introduces a universal categorical construction linking reversible dynamics on open systems to arbitrary dynamics on closed systems, unifying quantum channels and classical computing within a reversible framework.

Contribution

It presents a novel categorical completion that unifies quantum and classical dynamics, demonstrating their reversible foundations through a universal construction.

Findings

Unifies quantum channels and classical computing within a reversible categorical framework.

Shows how to 'undo' the construction, revealing reversible foundations of mixed quantum and classical theories.

Provides a new perspective on the relationship between open and closed system dynamics.

Abstract

We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the restriction affine completion of a monoidal restriction category quotiented by well-pointedness. This categorical completion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennett's method. Moreover, in these two cases, we show how our construction can be essentially 'undone' by a further universal construction. This shows how both mixed quantum theory and classical computation rest on entirely reversible foundations.