A causal derivation of the algebraic approach to quantum systems

TL;DR

This paper introduces a causal perspective on quantum systems, deriving the algebraic operator framework from causal structures and clarifying the nature of classical-quantum systems within this approach.

Contribution

It provides a causal derivation of the algebraic formalism of quantum systems, linking causal structures to von Neumann algebras and extending the framework to classical-quantum systems.

Findings

Every quantum system corresponds to a unique von Neumann algebra.

Classical-quantum systems are represented by commutative von Neumann algebras.

The causal view contrasts with the epistemic view, which is incompatible with the algebraic approach.

Abstract

It is commonly assumed that every quantum system is represented by some algebra of operators. Doubt is cast on this assumption by what appears, at first glance, to be a reasonable candidate for a quantum system that is not naturally represented by any algebra. To resolve this puzzle, this work draws inspiration from recent frameworks for causal modelling in quantum theory to propose a "causal view" of quantum systems. The causal view defines quantum systems purely in terms of the causal structure of the unitary dynamics. The algebraic representation of quantum systems is derived from the causal view: it is proven that every quantum system corresponds to a unique von Neumann algebra of operators. The causal view is extended with a definition of a "classical quantum system" inspired by quantum Darwinism. It is shown that such a system corresponds to a unique commutative von Neumann operator algebra, completing the derivation of the traditional algebraic approach to quantum systems from the causal view. The causal view is contrasted with the "epistemic view" of quantum systems, which is incompatible with the algebraic approach.