Algebraic Approach to Entanglement and Entropy

TL;DR

This paper introduces an algebraic framework for quantum entanglement and entropy, emphasizing the role of observable algebras and states, which simplifies the analysis of subsystems and entanglement in systems of identical particles.

Contribution

It develops a general algebraic approach based on the Gelfand-Naimark-Segal construction, providing new tools for studying entanglement and entropy without relying on Hilbert space as a fundamental concept.

Findings

Proposes a new entanglement measure applicable to various particle statistics.

Overcomes partial trace issues in identical particle systems.

Analyzes subsystem dynamics via subalgebra restrictions.

Abstract

We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gelfand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed, that can be applied to systems of particles obeying Fermi, Bose, para- and even braid group statistics. The generality of the method is also illustrated by the study of time evolution of subsystems emerging from restriction to subalgebras. Also, problems related to anomalies and quantum epistemology are discussed.