Bell states for fermions in loop quantum gravity

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.