Bell correlations at finite temperature

Matteo Fadel1 and Jordi Tura2

1Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

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We show that spin systems with infinite-range interactions can violate at thermal equilibrium a multipartite Bell inequality, up to a finite critical temperature $T_c$. Our framework can be applied to a wide class of spin systems and Bell inequalities, to study whether nonlocality occurs naturally in quantum many-body systems close to the ground state. Moreover, we also show that the low-energy spectrum of the Bell operator associated to such systems can be well approximated by the one of a quantum harmonic oscillator, and that spin-squeezed states are optimal in displaying Bell correlations for such Bell inequalities.

Measurements on quantum-mechanical systems can display statistics that escape our intuition, developed by our every-day interaction with a "classical" world. In 1964, John Bell developed a mathematical framework (Bell experiment) that allows to distinguish between statistics that are compatible with our intuitive view of the world (local realism) and those that go beyond, which we only know how to produce using intrinsically quantum effects (Bell correlations). Observing Bell correlations in systems that are composed of few particles is possible by performing conceptually simple, albeit technically challenging, Bell experiments. The case of multipartite systems naturally presents many more challenges, both theoretically and experimentally. Recently, however, important developments in the field allowed for the experimental detection of Bell correlations in many-body systems composed of hundreds and hundreds of thousands of atoms. Apart from these tailored experiments, where Bell correlations were prepared by artificially controlling the interactions among the atoms, we may ask whether Bell correlations can naturally occur in many-body systems. Theoretically, it has been proven that the ground state of some Hamiltonians show Bell correlations, but in practice a system can never be prepared exactly in its ground state, leaving the previous question open. Here, after showing that Bell correlations occur in the ground state of a physically relevant class of spin Hamiltonians, we show that they persist at finite temperature up to a critical value. For reasonable experimental parameters, we foresee the possibility of preparing Bell-correlated many-body spin systems by just cooling their spin degree of freedom to few pico Kelvins. As a result of our analysis, we also show that these low-energy states can be well-approximated by an important class of quantum-mechanical states, called spin-squeezed states.

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