  Quantum electronics is significantly involved in the development of the field
of quantum information processing. In this domain, the growth of Blind Quantum
Source Separation and Blind Quantum Process Tomography has led, within the
formalism of the Hilbert space, to the introduction of the concept of a
Random-Coefficient Pure State, or RCPS: the coefficients of its development in
the chosen basis are random variables. This paper first describes an
experimental situation necessitating its introduction. While the von Neumann
approach to a statistical mixture considers statistical properties of an
observable, in the presence of an RCPS one has to manipulate statistical
properties of probabilities of measurement outcomes, these probabilities then
being themselves random variables. It is recalled that, in the presence of a
von Neumann statistical mixture, the consistency of the density operator \r{ho}
formalism is based on a postulate. The interest of the RCPS concept is
presented in the simple case of a spin 1/2, through two instances. The most
frequent use of the \r{ho} formalism by users of quantum mechanics is a
motivation for establishing some links between a given RCPS and the language of
the density operator formalism, while keeping in mind that the situation
described by an RCPS is different from the one which has led to the
introduction of \r{ho}. It is established that the Landau - Feynman use of
\r{ho} is mobilized in a situation differing from both the von Neumann
statistical mixture and the RCPS. It is shown that the use of the higher-order
moments of a well-chosen random variable helps solving a problem already
identified by Zeh in 1970.
