Separable States Are More Disordered Globally than Locally

M. A. Nielsen; J. Kempe

doi:10.1103/PhysRevLett.86.5184

Separable States Are More Disordered Globally than Locally

M. A. Nielsen and J. Kempe

Phys. Rev. Lett. 86, 5184 – Published 28 May 2001

Abstract

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.

Received 29 November 2000

DOI:https://doi.org/10.1103/PhysRevLett.86.5184

Authors & Affiliations

M. A. Nielsen^1,* and J. Kempe^2,†

¹Centre for Quantum Computer Technology, University of Queensland, Queensland 4072, Australia
²Departments of Mathematics and Chemistry, University of California, Berkeley, California 94270

^*Email address: nielsen@physics.uq.edu.au
^†Also at École Nationale Superieure des Télécommunications, Paris, France. Email address: kempe@math.berkeley.edu