  Many constructions of randomness extractors are known to work in the presence
of quantum side information, but there also exist extractors which do not
[Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors $\psi$
with a bound on the second largest eigenvalue
$\lambda_{2}(\psi^{\dagger}\circ\psi)$ are quantum-proof. We then discuss fully
quantum extractors and call constructions that also work in the presence of
quantum correlations decoupling. As in the classical case we show that spectral
extractors are decoupling. The drawback of classical and quantum spectral
extractors is that they always have a long seed, whereas there exist classical
extractors with exponentially smaller seed size. For the quantum case, we show
that there exists an extractor with extremely short seed size
$d=O(\log(1/\epsilon))$, where $\epsilon>0$ denotes the quality of the
randomness. In contrast to the classical case this is independent of the input
size and min-entropy and matches the simple lower bound
$d\geq\log(1/\epsilon)$.
