An effect-theoretic reconstruction of quantum theory

TL;DR

This paper reconstructs quantum theory by deriving it from physically meaningful assumptions within effectus theory, showing that such theories embed into categories of C*-algebras, thus providing a foundational justification for their use.

Contribution

It introduces a new reconstruction of quantum theory based on effectus theory, avoiding convex structures and tensor products, and characterizes theories that embed into C*-algebras.

Findings

Finite-dimensional PETs embed into Euclidean Jordan algebras.

Monoidal PETs embed into categories of real or complex C*-algebras.

The approach provides a physically motivated foundation for operator algebras in quantum theory.

Abstract

An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C*-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory. In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect. We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C*-algebras, which completes our reconstruction.