  Recently, it has been argued that quantum mechanics is a complete theory, and
that different quantum states do necessarily correspond to different elements
of reality, under the assumptions that quantum mechanics is correct and that
measurement settings can be freely chosen. In this work, we prove that this
result is a consequence of an unnecessarily strong mathematical expression of
the free choice assumption, which embodies more conditions than explicitly
stated. The issues of the completeness of quantum mechanics, and of the
interpretation of the state vector, are by no means resolved. Taking this
perspective, we describe how the recently introduced class of crypto-nonlocal
hidden variables theories can be used to characterize the maximal possible
departure from quantum mechanics, when the system consists of a pair of qubits.
