Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions (...) are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems. (shrink)

The immediate motive occasioning this letter was the worker E. Kopanitsyn's letter to the newspaper Izvestiia in which he eloquently recounts his difficulties with a new machine manufactured in a factory in the city of Cheliabinsk. After telling what effort it had cost him to get this machine going, Kopanitsyn in a fit of temper exclaims: "I am completely unable to understand one simple thing: How could such a machine come into being at all? The hands of many workers touched (...) it, except that these were not working hands. Can it really be true that no one, not a single person, was aroused enough to say: Just what are we doing here, brothers?". (shrink)