Motivation

One of the main goals of quantum foundations is to find a set of reasonable, physical or informatic axioms which uniquely pick out quantum theory from the ‘space of all possible theories’.

It is problematic to consider all theories so instead some underlying framework of theories is used to study some class of theories, the most successful of these to date is that of general probabilistic theories (GPTs).

Categorical quantum mechanics sits inside another, more general, framework, that of process theories. This is essentially just another name for monoidal categories where a physical interpretation is made of the objects and morphisms in the category. i.e. objects correspond to physical objects and morphisms to the possible processes between said objects. This is a new perspective on "theory space" as it puts composition of systems at its heart avoiding problems endemic to frameworks such as quantum logic.

In the current presentation of categorical quantum mechanics the process theory framework is physically motivated from an operational perspective by considering what scientists could actually do in a lab. Categorical composition of morphisms corresponds to `doing one thing and then another’ to the same system and the monoidal composition to `doing two things at once’ to two separate systems. However to pick out quantum theory there are a series of ad hoc choices to be made, such as defining scalars to be the complex numbers, and allowing for addition of morphisms. These are not physically motivated instead they are motivated by the mathematical structure of standard quantum theory.

Open Problem

A current objective is therefore to find a set of physically motivated axioms which will reproduce this mathematical structure. Two possible routes to this are to take an axiomatisation in the GPT framework and try to find a more categorical approach (see Notions of Purity) and the other is to try to work from the ground up and to consider axioms that might be more natural from the categorical perspective.

Related Work

For the current “non-physically motivated” set up of categorical quantum mechanics, *need an updated reference for this*,

is a good starting point.

Reconstructions of quantum theory by G. Chiribella, G. M. D'Ariano, P. Perinotti,

and L. Hardy,

both rely on a diagrammatic framework in their reconstruction, and although they are not explicitly categorical it is there in spirit.