Generalization of Gisin's Theorem to Quantum Fields

TL;DR

This paper extends Gisin's theorem to quantum fields, demonstrating that any pure state with mode entanglement, including states with undefined particle number, violates Bell inequalities, thus linking entanglement with non-classical correlations.

Contribution

The authors generalize Gisin's theorem to quantum fields with indefinite particle number, providing a method to construct Bell inequality violations for such states.

Findings

Pure states with mode entanglement violate Bell inequalities.

The generalization applies to states with infinite particles.

Experimental feasibility of Bell tests with linear optics is discussed.

Abstract

We generalize Gisin's theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser-Horne inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the Clauser-Horne inequality. We also show that our generalization of Gisin's theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.