Probability distributions consistent with a mixed state

For a fixed density matrix it is natural to ask what class of ensembles $pi ,uc i&% gives rise to that density matrix? This problem was addressed by Schrödinger @3#, whose results have been extended by Jaynes @4#, and by Hughston, Jozsa, and Wootters @5#. The result of these investigations, the classification theorem for ensembles, has been of considerable utility in quantum statistical mechanics, quantum information theory, quantum computation, and quantum error correction. In this paper we use the classification theorem for ensembles to obtain an explicit classification of probability distributions (pi) such that there exist pure states uc i& satisfying r5( ipiuc i&^c iu, for some fixed density matrix r . This is done in Sec. II. Section III illustrates the result with several simple applications to quantum mechanics and quantum information theory. Section IV concludes the paper.