A Bestiary of Sets and Relations

TL;DR

This paper explores the categorical structure of quantum mechanics within the finite sets and relations framework, revealing similarities and differences with traditional quantum categories and uncovering exotic features.

Contribution

It provides a detailed categorical analysis of quantum mechanics in fRel, highlighting structural similarities with fdHilb and demonstrating that fRel is a local category with unique properties.

Findings

fRel shares operational features with fdHilb

fRel is proven to be a local category

The category exhibits exotic structural phenomena

Abstract

Building on established literature and recent developments in the graph-theoretic characterisation of its CPM category, we provide a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations. On the way, we highlight the wealth of exotic beasts that hide amongst the extensive operational and structural similarities that the theory shares with more traditional arenas of categorical quantum mechanics, such as the category fdHilb. We conclude our journey by proving that fRel is local, but not without some unexpected twists.