A computer scientist's reconstruction of quantum theory

TL;DR

This paper presents a novel categorical reconstruction of quantum theory that accommodates infinite-dimensional systems, classical and mixed states, without relying on real or complex number structures, from a computer science perspective.

Contribution

It introduces a compositional, effectus-theoretic framework for quantum theory that includes infinite dimensions and mixed states, without assuming real or complex number structures.

Findings

Includes no restrictions on system dimension

Supports classical, quantum, and mixed systems

Uses category theory for a foundational reconstruction

Abstract

The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure of quantum systems comes from. From a computer scientist's perspective we would like to study quantum theory in a way that allows interesting transformations and compositions of systems and that also includes infinite-dimensional datatypes. Here we present such a compositional reconstruction of quantum theory that includes infinite-dimensional systems. This reconstruction is noteworthy for three reasons: it is only one of a few that includes no restrictions on the dimension of a system; it allows for both classical, quantum, and mixed systems; and it makes no a priori reference to the structure of the real (or complex) numbers. This last point is possible because we frame our results in the language of category theory, specifically the categorical framework of effectus theory.