  We discuss the discriminating power of separability criteria, which are based
on the spectrum of a quantum state and its reductions. Common examples are
entropic inequalities utilizing conditional Tsallis or Renyi entropies. We
prove that these inequalities are implied by the reduction criterion for any
positive value of the entropic parameters. We show however, that arbitrary sets
of criteria based on spectral and local information can never be sufficient by
establishing a separable, isospectral and locally undistinguishable counterpart
for any Werner state in odd dimensions. For the case of two qubit systems we
show that a simple controlled phase gate operation can produce an isospectral,
entangled state out of a separable one, which has the same reductions.
