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 glo
bally. 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 nonsepa
rable. 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 s
tates in arbitrary dimensions.