Energy as a Detector of Nonlocality of Many-Body Spin Systems

Abstract

We present a method to show that low-energy states of quantum many-body interacting systems in one spatial dimension are nonlocal. We assign a Bell inequality to the Hamiltonian of the system in a natural way and we efficiently find its classical bound using dynamic programming. The Bell inequality is such that its quantum value for a given state, and for appropriate observables, corresponds to the energy of the state. Thus, the presence of nonlocal correlations can be certified for states of low enough energy. The method can also be used to optimize certain Bell inequalities: in the translationally invariant (TI) case, we provide an exponentially faster computation of the classical bound and analytically closed expressions of the quantum value for appropriate observables and Hamiltonians. The power and generality of our method is illustrated through four representative examples: a tight TI inequality for 8 parties, a quasi TI uniparametric inequality for any even number of parties, ground states of spin-glass systems, and a non-integrable interacting XXZ-like Hamiltonian. Our work opens the possibility for the use of low-energy states of commonly studied Hamiltonians as multipartite resources for quantum information protocols that require nonlocality.

Bipartite nonlocality with a many-body system

“…Exceptions have been found recently. In particular [2][3][4], showed that some multi-partite states are able to violate Bell inequalities using the product of two local observables. These inequalities paved the way for Bell tests with many-body systems where few-body correlations-correlations between a small number of local observables-are accessible only.…”

“…This requirement have been lifted with the use of a Bell correlation witness (see below for details) which were used to demonstrate the presence of Bell correlations in many-body states [5][6][7]. This implies that if one could measure the constituent particle individually, then the observed correlations are strong enough to violate Bell inequalities presented in [2][3][4]. Such an individual addressing is unrealistic for a large number of particles.…”

Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing

“…Indeed, any method that is based on the full information about either the state or the observed correlations is bound to become intractable already for medium-large systems, since such information scales exponentially with the number of particles involved. Interestingly, it has already been shown that nonlocality can be assessed with the knowledge of few-body correlations only [19][20][21][22][23], which require generally a polynomial scaling number of measurements to be estimated. Moreover, Bell inequalities that consist of a constant amount of terms have also been introduced [19,20], opening the way to the first experimental detections of Bell correlations in many-body systems of hundreds [24] and hundreds of thousands of atoms [25].…”

Bell Correlations in a Many-Body System with Finite Statistics

“…Similar inequalities were also obtained for translationally invariant systems [23], or based on Hamiltonians [24]. Here, we derive a similar family of Bell inequalities that is invariant under arbitrary permutations of parties but allows for an arbitrary number of measurement settings per party.…”