  Ontological theories of quantum mechanics provide a realistic description of
single systems by means of well-defined quantities conditioning the measurement
outcomes. In order to be complete, they should also fulfil the minimal
condition of macroscopic realism. Under the assumption of outcome determinism
and for Hilbert space dimension greater than two, they were all proved to be
contextual for projective measurements. In the recent years a generalized
concept of non-contextuality was introduced that applies also to the case of
outcome indeterminism and unsharp measurements. It was pointed out that the
Beltrametti-Bugajski model is an example of measurement non-contextual
indeterminist theory. Here we provide a simple proof that this model is the
only one with such a feature for projective measurements and Hilbert space
dimension greater than two. As a corollary, von Neumann measurement
non-contextuality implies non-contextuality for unsharp measurements. By noting
that the Beltrametti-Bugajski model does not satisfy the condition of
macroscopic realism, we arrive at the conclusion that the only way to solve the
measurement problem in the framework of an ontological theory is relaxing the
hypothesis of non-contextuality in its generalized sense.

Quantum Physics