  Given a list of n complex numbers, when can it be the spectrum of a quantum
channel, i.e., a completely positive trace preserving map? We provide an
explicit solution for the n=4 case and show that in general the
characterization of the non-zero part of the spectrum can essentially be given
in terms of its classical counterpart - the non-zero spectrum of a stochastic
matrix. A detailed comparison between the classical and quantum case is given.
We discuss applications of our findings in the analysis of time-series and
correlation functions and provide a general characterization of the peripheral
spectrum, i.e., the set of eigenvalues of modulus one. We show that while the
peripheral eigen-system has the same structure for all Schwarz maps, the
constraints imposed on the rest of the spectrum change immediately if one
departs from complete positivity.
