Quantum Ensembles and the Statistical Operator: A Tutorial

The main purpose of this tutorial is to elucidate in details what should be meant by ensemble of states in quantum mechanics, and to properly address the problem of discriminating, exactly or approximately, two diÆerent ensembles. To this aim we review the notion and the definition of quantum ensemble as well as its relationships with the concept of statistical operator in quantum mechanics. We point out the implicit assumptions contained in introducing a correspondence between quantum ensembles and the corresponding to singleparticle statistical operator, and discuss some issues arising when these assumptions are not satisfied. We review some subtleties leading to apparent paradoxes, and illustrate the role of approximate quantum cloning. In particular, we review some examples of practical interest where diÆerent (but equivalent) preparations of a quantum system, i.e. diÆerent ensembles corresponding to the same single-particle statistical operator, may be successfully discriminated exploiting multiparticle correlations, or some a priori knowledge about the number of particles in the ensemble.