Generalization of Gisin’s theorem to quantum fields

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 the 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 the constructed Bell test and provide a necessary condition for the realizability of this test within the realm of passive linear optics.

Open dynamics of entanglement in mesoscopic bosonic systems

“…One of the crucial quantum phenomena is entanglement, which is one of the core resources in many quantum information tasks. This is even more so since in the quantum field theory the entanglement of pure states is fundamentally associated with Bell non-classicality [25], which is necessary for device-independent quantum key distribution. For such tasks, particularly compelling is the encoding of information in light modes, as light can be quickly distributed over long distances [16].…”