Preserving entanglement under decoherence and sandwiching all separable states

Abstract

Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles increases, the distance between two of the hyperplanes which sandwich the separable states goes to zero. It is easy to decide if a state Q is in the neighborhood of entangled states we construct for an entangled state E. One merely has to check if the trace of EQ is greater than a constant which depends upon E and which we determine.