Solving the Zeh problem about the density operator with higher-order statistics

TL;DR

This paper challenges the traditional view that identical density operators represent the same statistical mixture, showing that higher-order statistics can distinguish different mixtures, thus refining the interpretation of density operators in quantum mechanics.

Contribution

It demonstrates that different mixtures with the same density operator can be distinguished using higher-order statistics, addressing a long-standing controversy in quantum information theory.

Findings

Density operators do not always uniquely specify statistical mixtures.

Higher-order statistics can differentiate mixtures with identical density operators.

The results clarify the interpretation of density operators in quantum mechanics.

Abstract

Since a 1932 work from von Neumann, it is generally considered that if two statistical mixtures are represented by the same density operator \r{ho}, they should in fact be considered as the same mixture. In a 1970 paper, Zeh, considering this result to be a consequence of what he called the measurement axiom, introduced a thought experiment with neutron spins and showed 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 this paper, stimulated by our previous works in the field of Quantum Information Processing, we show that the two mixtures imagined by Zeh, with the same \r{ho}, should however be distinguished. We show that this result suppresses a restriction unduly installed on statistical mixtures, but does not affect the general use of \r{ho}, e.g. in quantum statistical mechanics, and the von Neumann entropy keeps its own interest and even helps clarifying this confusing consequence of the measurement axiom. In order to avoid any ambiguity, the identification of the introduction of this postulate, which von Neumann rather suggested to be a general property, is given in an appendix where it is shown that Zeh was right when he spoke of a measurement axiom and identified his problem. The use and content of a density operator is also discussed in another physical case which we are led to call the Landau-Feynman situation, and which implies the concept of entanglement rather than the one of mixed states.