Bell states for fermions in loop quantum gravity

TL;DR

This paper explores how fermionic entanglement, especially Bell states, can be characterized within loop quantum gravity, revealing subtle issues and proposing a new observable that demonstrates Bell inequality violations.

Contribution

It introduces a novel approach to defining fermionic entanglement in loop quantum gravity and demonstrates Bell inequality violations with coupled fermion-geometry states.

Findings

Fermionic entanglement in loop quantum gravity is subtle and challenging to define.

A new observable related to fermion spin normal components is proposed.

States coupling fermions and quantum geometry can violate Bell-CHSH inequality.

Abstract

Fermion fields are fundamental for the description of nature and also fit very naturally into the framework of loop quantum gravity. Motivated partially by proposals to use gravitationally mediated entanglement of matter as a witness for the quantum nature of gravity, we investigate how such entanglement can be defined and investigated in loop quantum gravity. In particular, we ask how a pair of fermions in a Bell state could be described in loop quantum, gravity. We demonstrate that the notion of fermionic entanglement in loop quantum gravity is subtle, by showing that some potential ways to define it fail. We then investigate a kinematical observable involving both, fermionic and gravitational degrees of freedom, the component of the fermion spin normal to a surface. We study its properties, and compare it to the standard operator for components of spin in a given direction in quantum mechanics. Using these normal components of spin, we define a kinematical observable that measures the correlation between space-like separated fermions which closely mirrors the CHSH observable. Finally, we exhibit states of the fermions coupled to quantum geometry that violate the Bell-CHSH inequality.