  We analyze and compare the mathematical formulations of the criterion for
separability for bipartite density matrices and the Bell inequalities. We show
that a violation of a Bell inequality can formally be expressed as a witness
for entanglement. We also show how the criterion for separability and a
description of the state by a local hidden variable theory, become equivalent
when we restrict the set of local hidden variable theories to the domain of
quantum mechanics. This analysis sheds light on the essential difference
between the two criteria and may help us in understanding whether there exist
entangled states for which the statistics of the outcomes of all possible local
measurements can be described by a local hidden variable theory.
