Additivity and non-additivity of multipartite entanglement measures

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. Firstly, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interest have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state and the generalization of Dür’s multipartite bound entangled states. We also prove the additivity of the other two measures for some of these examples. Secondly, we show the nonadditivity of GM of all antisymmetric states of three or more parties. We also provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In addition, we derive analytical formulae for the three measures of one copy and two copies of the antisymmetric projector states, respectively. Thirdly, we show, with a statistical approach, that almost all multipartite pure states with a sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement, and they have non-additive GM. Hence, more states may be suitable for universal quantum computation if measurements can be performed on two copies of the resource states. We also show that almost all the multipartite pure states cannot 4 Author to whom any correspondence should be addressed. New Journal of Physics 12 (2010) 083002 1367-2630/10/083002+41$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft