Reconstructing quantum theory from diagrammatic postulates

TL;DR

This paper reconstructs finite-dimensional quantum theory using diagrammatic, category-theoretic, and process-theoretic postulates, introducing symmetric purification and exploring classical-quantum interactions.

Contribution

It introduces symmetric purification as a novel postulate applicable to both classical and quantum theories, and provides a diagrammatic framework for reconstructing quantum theory.

Findings

Reconstruction of quantum theory from diagrammatic postulates

Introduction of symmetric purification applicable to classical and quantum theories

Diagrammatic presentation of generalised probabilistic theories

Abstract

We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the `fully quantum' systems as the final step. We propose two novel alternative manners of doing so, `no-leaking' (roughly that information gain causes disturbance) and `purity of cups' (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the `sharp dagger' is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.