Categorical Quantum Mechanics II: Classical-Quantum Interaction

TL;DR

This paper develops a category-theoretic framework for classical-quantum interaction, describing systems, processes, and entanglement, and introduces new concepts like phase spiders and genuine quantumness.

Contribution

It extends categorical quantum mechanics to include classical-quantum interaction, defining classicality, mixedness, entanglement, and quantum protocols within a unified process-theoretic approach.

Findings

Classicality is represented by 'spiders' that fuse when connected.

Pure processes are extremal in the process space.

GHZ and W states are characterized as spider-like processes.

Abstract

This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part we focus on classical-quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioural properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by 'spiders' which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully-comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of 'genuine quantumness', from which special processes called 'phase spiders' arise, and get a first glimpse of quantum non-locality.