  The Kochen-Specker theorem, Bell inequalities, and several other tests that
were designed to rule out hidden-variable theories, assume the existence of
observables having infinitely sharp eigenvalues. A paradigmatic example is
spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided
into two classes, spin-up and spin-down, in correspondence to the two spots
observed on a detection screen. The spot's finite size is attributed to
imperfections of the measuring device. This assumption turns the experimental
output into a dichotomic, discrete one, thereby allowing the assignment of each
spot to an infinitely sharp eigenvalue. Alternatively, one can assume that the
spot's finite size stems from eigenvalues spanning a continuous range. Can we
disprove such an assumption? Can we rule out hidden-variable theories that
reproduce quantum predictions by assuming that, e.g., the electron's magnetic
moment is not exactly the same for all electrons? We address these questions by
focusing on the Peres-Mermin version of the Bell-Kochen-Specker theorem. It is
shown that the assumption of unsharp eigenvalues precludes ruling out
non-contextual hidden-variable theories and hence quantum contextuality does
not arise. Analogous results hold for Bell-like inequalities. This represents a
new loophole that spoils several fundamental tests of quantum mechanics and
issues the challenge to close it.

Quantum Physics