Separable States Are More Disordered Globally than Locally

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 nonseparable. 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.