A Schmidt number for density matrices

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. Similar as the positive map criterion for entanglement, we show that k-positive maps witness Schmidt number. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number {\it does not necessarily increase} when taking tensorcopies of a density matrix $\rho$; we give an example of a density matrix for which the Schmidt number of $\rho$ and $\rho \otimes \rho$ is both 2.