Solving the Zeh Problem About the Density Operator with Higher-Order Statistics

Since a 1932 work from von Neumann, it has been considered that if two statistical mixtures are represented by the same density operator rho, they should, in fact, be considered as the same mixture. In a 1970 paper, Zeh introduced a thought experiment with neutron spins, and suggested that, in that experiment, the density operator could not tell the whole story. Since then, no consensus has emerged yet, and controversies on the subject still presently develop. In his 1995 book, speaking of the use of the density operator, Peres spoke of a von Neumann postulate. In this paper, keeping the random variable used by von Neumann in his treatment of statistical mixtures, but also considering higher-order moments of this random variable, it is established that the two mixtures imagined by Zeh, with the same rho, should however be distinguished. We show that the rejection of that postulate, installed on statistical mixtures for historical reasons, does not affect the general use of rho, e.g., in quantum statistical mechanics, and the von Neumann entropy keeps its own interest and even helps clarifying that confusing consequence of the postulate identified by Peres.