Entanglement and algebraic independence in fermion systems

TL;DR

This paper discusses the algebraic formulation of entanglement in fermion systems, emphasizing the importance of micro-causality over algebraic independence for a consistent theoretical framework.

Contribution

It clarifies the distinction between micro-causality and algebraic independence in fermion systems and advocates for a formulation aligned with micro-causality.

Findings

Algebraic independence and micro-causality are distinct in fermion systems.

A consistent entanglement framework should prioritize micro-causality.

The approach differs from the bosonic case due to fermions' anti-commutation relations.

Abstract

In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.