The aim of quantum foundations can broadly be said to be to understand why quantum theory is as it is, what we can do with it, and how to best describe it. The methods used for this are varied ranging from almost pure mathematics to philosophy to experimental physics.

There are many different approaches to this, on the more traditional side the categorical approach has leaked into quantum informatic reformulations such as those by Chiribella, D'Ariano and Perinotti and another by L. Hardy both very much influenced by the use of string diagrams by Abramsky and Coecke. Coecke et al. pursue a diagrammatic process theory approach to both reformulating quantum theory and understanding concepts (such as no-signalling, causality and quantum information protocols) from a categorical/process theory perspective. Abramsky et al. are using sheaf-theoretic tools to explore contextuality and non-locality in quantum theory.

Historic/Current research

The main successes to date include:

The categorical reconstruction of quantum theory provides intuitive diagrammatic representations of quantum processes such as teleportation and entanglement sharing protocols.

A new view of Spekkens toy model as embedded in FRel leading to a comparison of this to stabiliser quantum theory highlighting the phase group as the fundamental difference between the theories. This difference must therefore be the element that leads to non-locality in stabiliser theory.

More clearly understanding the link between terminality of the monoidal unit, causality and signalling in process theories.

Using sheaf theory to gain a new perspective on non-locality and contextuality in quantum theory.

Other work which is not strictly using the categorical framework but inspired by the diagrammatic reasoning includes two reconstructions of quantum theory from informational axioms, these both a diagrammatic framework at the centre of their reconstructions.

Resource theories have been instrumental in understanding quantum theory, in particular the structure of entanglement. A general view of resource theories by Coecke, Fritz and Spekkens uses symmetric monoidal categories as the underlying framework.

Future directions of research

Current open problems are