  For several types of correlations: mixed-state entanglement in systems of
distinguishable particles, particle entanglement in systems of
indistinguishable bosons and fermions and non-Gaussian correlations in
fermionic systems we estimate the fraction of non-correlated states among the
density matrices with the same spectra. We prove that for the purity exceeding
some critical value (depending on the considered problem) fraction of
non-correlated states tends to zero exponentially fast with the dimension of
the relevant Hilbert space. As a consequence a state randomly chosen from the
set of density matrices possessing the same spectra is asymptotically a
correlated one. To prove this we developed a systematic framework for detection
of correlations via nonlinear witnesses.
