  We study the problem of whether all bipartite quantum states having a
prescribed spectrum remain positive under the reduction map applied to one
subsystem. We provide necessary and sufficient conditions, in the form of a
family of linear inequalities, which the spectrum has to verify. Our conditions
become explicit when one of the two subsystems is a qubit, as well as for
further sets of states. Finally, we introduce a family of simple entanglement
criteria for spectra, closely related to the reduction and positive partial
transpose criteria, which also provide new insight into the set of spectra that
guarantee separability or positivity of the partial transpose.
