  We consider the inverse eigenvalue problem for entanglement witnesses, which
asks for a characterization of their possible spectra (or equivalently, of the
possible spectra resulting from positive linear maps of matrices). We
completely solve this problem in the two-qubit case and we derive a large
family of new necessary conditions on the spectra in arbitrary dimensions. We
also establish a natural duality relationship with the set of absolutely
separable states, and we completely characterize witnesses (i.e., separating
hyperplanes) of that set when one of the local dimensions is 2.
