  It is generally expected that decoherence processes will erase the quantum
properties of the inflationary primordial spectra. However, given the weakness
of gravitational interactions, one might end up with a distribution which is
only partially decohered. Below a certain critical change, we show that the
inflationary distribution retains quantum properties. We identify four of
these: a squeezed spread in some direction of phase space, non-vanishing
off-diagonal matrix elements, and two properties used in quantum optics called
non-$P$-representability and non-separability. The last two are necessary
conditions to violate Bell's inequalities. The critical value above which all
these properties are lost is associated to the `grain' of coherent states. The
corresponding value of the entropy is equal to half the maximal (thermal)
value. Moreover it coincides with the entropy of the effective distribution
obtained by neglecting the decaying modes. By considering backreaction effects,
we also provide an upper bound for this entropy at the onset of the adiabatic
era.
