Quantum Ensembles and the Statistical Operator: A Tutorial
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Yinxiμu Zhan,Matteo G. A. Paris. Quantum Ensembles and the Statistical Operator: A Tutorial. International Journal of Software and Informatics, 2014,8(3-4):241~253
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Fund:This work has been supported by MIUR through the FIRB project LiCHIS (grant RBFR10YQ3H), by EU through the Collaborative Projects QuProCS (Grant Agreement 641277) and by UniMI through the H2020 Transition Grant 15-6-3008000-625.
Abstract: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 different 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 single-particle 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 different (but equivalent) preparations of a quantum system, i.e. different 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.
keywords:quantum ensembles  statistical operator  density matrix  no-cloning theorem
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