Universal Properties of Partial Quantum Maps

TL;DR

This paper introduces a universal categorical construction for finite-dimensional quantum and classical computations, enabling better understanding and design of quantum programming languages.

Contribution

It provides a universal construction of the category of quantum maps from the category of Hilbert spaces, applicable to any dagger rig category, with implications for quantum programming language semantics.

Findings

Constructs a universal category for quantum maps with dilation theory.

Captures hybrid quantum/classical computation with nontermination.

Facilitates the design of quantum programming languages.

Abstract

We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be applied to any dagger rig category, is described in three steps, each associated with their own universal property, and draws on results from dilation theory in finite dimension. In this way, we explicitly construct the category that captures hybrid quantum/classical computation with possible nontermination from the category of its reversible foundations. We discuss how this construction can be used in the design and semantics of quantum programming languages.