Local Operations and Field Mediated Entanglement without a Local Tensor Product Structure

TL;DR

This paper develops a framework for understanding entanglement and locality in gauge theories without relying on a traditional tensor product structure, applying it to a lattice electromagnetism model and implications for quantum gravity tests.

Contribution

It introduces gauge-invariant local algebras and a Hilbert space decomposition for gauge theories, enabling quantum information concepts without a local tensor product structure.

Findings

Discretized electromagnetism satisfies an LOCC-like theorem.

Entanglement requires genuine quantum field interactions.

Framework applicable to quantum gravity tests.

Abstract

Quantum information has become a powerful tool for probing the structure of quantum field theories, yet its application to gauge theories remains subtle. On the one hand, quantum information theory assumes subsystem locality, i.e.~the factorization of the total Hilbert space into subsystems. On the other hand, gauge constraints prevent the total Hilbert space to decompose into a spacetime-local tensor product structure. Because the Hilbert space structure of gauge theories does not accommodate the subsystem decomposition used in quantum information theory, standard information-theoretic results, such as the Local Operations and Classical Communication (LOCC) theorem, cannot be used straightforwardly in the context of gauge theories. In this work, we bridge this gap in the case of a two-dimensional lattice gauge model that captures key features of electromagnetism. In particular, we construct gauge-invariant local algebras and derive a physically meaningful decomposition of the Hilbert space, providing an operationally consistent notion of locality in the absence of a local tensor-product structure. We apply this framework to field-mediated entanglement protocols relevant to proposed tests of the quantum nature of gravity. We show that the discretized version of electromagnetism satisfies an analogue of the LOCC theorem: entanglement cannot be generated without genuine quantum field interactions, even in the absence of a spacetime-local tensor product factorization of the Hilbert space. This may point towards an operational way to define a subsystem structure for gauge theories.