Random-coefficient pure states, the density operator formalism and the Zeh problem

TL;DR

This paper introduces the concept of Random-Coefficient Pure States (RCPS) in quantum mechanics, linking it to the density operator formalism and addressing the Zeh problem through higher-order moments analysis.

Contribution

It presents the RCPS concept, explores its relation to the density operator formalism, and applies it to the Zeh problem using higher-order moments.

Findings

RCPS formalism describes quantum states with random coefficients.

Connections established between RCPS and density operator formalism.

Higher-order moments help solve the Zeh problem.

Abstract

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.