Schmidt number for density matrices - art. no. 040301

We introduce the notion of a Schmidt number of a bipartite density matrix. We show that k-positive maps witness the Schmidt number, in the same way th at positive maps witness entanglement. We determine the Schmidt number of t he family of states that is made from mixing the completely mixed state and a maximally entangled state. We show that the Schmidt number does not nece ssarily increase when taking tensor copies of a density matrix rho; we give an example of a density matrix for which the Schmidt numbers of rho and rh o x rho are both 2.