Bounds on detection of Bell correlations with entangled ultra-cold atoms in optical lattices under occupation defects

Bell non-locality stems from quantum correlations effectively identified using inequalities. Spin chains, simulated with ultra-cold atoms in optical lattices, Rydberg atoms in tweezer arrays, trapped ions, or molecules, allow single-spin control and measurement. Therefore, they are suitable for studying fundamental aspects of these correlations and non-locality. Occupation defects, such as vacancies or multiple atoms occupying a single site due to imperfect system preparation, limit the detection of Bell correlations. We investigate their impact using a simplified toy model parameterized by the probability of a site being singly occupied. We derive the corresponding Bell inequality and identify the smallest probability that establishes a lower bound for detecting Bell correlations. We relate the bound to two physical parameters leading to defects in occupations: non-zero temperature and filling factor, focusing on entangled ultra-cold atoms in optical lattices. Finally, we numerically validate the predictions of the toy model by full many-body simulations.