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

TL;DR

The paper tackles the challenge that gauge constraints prevent a local tensor-product Hilbert space in gauge theories, which complicates LOCC-based analyses of entanglement. It introduces a two-dimensional lattice toy model with gauge-invariant local algebras and two complementary Hilbert-space decompositions, enabling a sector-by-sector notion of locality and an operational LOCC framework. A generalized LOCC theorem is proven to apply within each sector, and a field-mediated entanglement protocol (FME) is explicitly realized in the toy model via gauge-invariant dressings that couple matter to the field, producing entanglement detectable on spins. This operational approach provides a concrete route to define subsystems and locality in gauge theories and points toward extensions to continuum QED and gravity, offering a groundwork for interpreting entanglement generation in gauge-field–mediated scenarios.

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.

Figures (10)

• Figure 1: Graphical representation of a portion of the lattice, with waves representing a non-trivial interaction between the sites.
• Figure 2: Representation of a portion of the region $A$.
• Figure 3: Graphical representation of the generators of $\mathcal{A}_A$ based on the Graphs notations introduced in Appendix \ref{['app:graph_def']}. The yellow circles and green rings label respectively the $\hat{p}$ and $\hat{b}$ operators at each site.
• Figure 4: Schematic representation of the FME protocol. Two massive particles, each in a spatial superposition, interact solely via the gravitational field while isolated from any other influence. The difference in distance between the interferometric paths is responsible for the generation of entanglement.
• Figure 5: Two square regions of dimension $M$, representing the laboratories in which the two parties $A$ and $B$ are located during a field-mediated entanglement (FME) experiment.