Partitions in quantum theory

TL;DR

This paper broadens the concept of system partitions in quantum theory to include non-factor sub-C* algebras, exploring their properties, physical interpretation, and representation via routed quantum circuits, revealing residual pseudo-nonlocality.

Contribution

It introduces a generalized definition of partitions into arbitrary sub-C* algebras, including non-factor ones, and analyzes their properties and physical implications.

Findings

Partitions can be represented by Hilbert space splitting using routed quantum circuits.

Some partitions exhibit residual pseudo-nonlocality, especially in fermionic systems.

The framework accommodates symmetry considerations in subsystem decomposition.

Abstract

Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In quantum theory, subsystems are usually framed as sub-C* algebras of the algebra of operators on the global system. However, most decompositional approaches have so far restricted their scope to the case of systems corresponding to factor algebras. We argue that this is a mistake: one should cater for the possibility for non-factor subsystems, arising for instance from symmetry considerations. Building on simple examples, we motivate and present a definition of partitions into an arbitrary number of parts, each of which is a possibly non-factor sub-C* algebra. We discuss its physical interpretation and study its properties, in particular with regards to the structure of algebras' centres. We prove that partitions, defined at the C*-algebraic level, can be represented in terms of a splitting of Hilbert spaces, using the framework of routed quantum circuits. For some partitions, however, such a representation necessarily retains a residual pseudo-nonlocality. We provide an example of this behaviour, given by the partition of a fermionic system into local modes.