  Bell inequalities were meant to test quantum mechanics vs local hidden
variable models, but can also be used to verify entanglement. For entanglement
verification purposes one assumes the validity of quantum mechanics as well as
quantum descriptions of one's measurements. With the help of these assumptions
it is possible to derive a strengthened Bell inequality whose violation implies
entanglement. We generalize known examples of such inequalities by relating the
expectation value of the Bell operator to a particular quantitative measure of
entanglement, namely the negativity. Moreover, we obtain statistics
illustrating the fact that violating a given (strengthened or not) Bell
inequality is a much more rare feat for a quantum state of two qubits than it
is to be entangled.
