  A remarkable feature of quantum entanglement is that an entangled state of
two parties, Alice (A) and Bob (B), may be more disordered locally than
globally. That is, S(A) > S(A,B), where S(.) is the von Neumann entropy. It is
known that satisfaction of this inequality implies that a state is
non-separable. In this paper we prove the stronger result that for separable
states the vector of eigenvalues of the density matrix of system AB is
majorized by the vector of eigenvalues of the density matrix of system A alone.
This gives a strong sense in which a separable state is more disordered
globally than locally and a new necessary condition for separability of
bipartite states in arbitrary dimensions. We also investigate the extent to
which these conditions are sufficient to characterize separability, exhibiting
examples that show separability cannot be characterized solely in terms of the
local and global spectra of a state. We apply our conditions to give a simple
proof that non-separable states exist sufficiently close to the completely
mixed state of $n$ qudits.
