Bell inequalities play a central rôle in the study of quantum non-locality and entanglement, with many applications in quantum information. Despite the huge literature on Bell inequalities, it is not easy to find a clear conceptual answer to what a Bell inequality is, or a clear guiding principle as to how they may be derived. In this paper, we introduce a notion of logical Bell inequality whic...

The Bell number B, is the number of partitions of a finite set whose cardinality is n. The modular properties of Bell numbers have been studied by many authors. It is well known that the sequence of the residues of Bell numbers modulo any integer satisfies a linear recurrence and is periodic [ 1, 68, 13, 143. When the modulus is prime, this recurrence is particularly simple. Touchard [16] shows...

A new decomposition form is introduced in this report to establish a criterion for the bi-partite separability of Bell diagonal states. A such criterion takes a quadratic inequality of the coefficients of a given Bell diagonal states and can be derived via a simple algorithmic calculation of its invariants. In addition, the criterion can be extended to a quantum system of higher dimension. Keyw...