Can differently prepared mixed states be distinguished?

A measurement has been proposed by d'Espagnat that would distinguish from one another some ensembles that are differently prepared but correspond to the same density matrix. Here, the idea is modified so that it becomes applicable to much more general situations. The method is illustrated in simple examples. Some matter of concern might then be that information could be transmitted by methods based on our idea in the EPR kind of experiments. A simple proof is given that this is impossible.