Proper Versus Improper Mixtures: Towards a Quaternionic Quantum Mechanics

TL;DR

This paper explores how proper and improper quantum mixtures can be distinguished using quaternionic quantum mechanics, providing a new mathematical framework to address the subentity problem in quantum theory.

Contribution

It introduces a quaternionic formulation that differentiates proper from improper mixtures, offering a novel mathematical approach to the subentity problem in quantum mechanics.

Findings

Proper and improper mixtures are represented by different density operators in quaternionic quantum mechanics.

The quaternionic framework allows for a clear mathematical distinction between proper and improper mixtures.

An example related to quantum measurement illustrates the practical implications of the approach.

Abstract

The density operators obtained by taking partial traces do not represent proper mixtures of the subsystems of a compound physical system, but improper mixtures, since the coefficients in the convex sums expressing them never bear the ignorance interpretation. As a consequence, assigning states to these subsystems is problematical in standard quantum mechanics (subentity problem). Basing on the proposal provided in the SR interpretation of quantum mechanics, where improper mixtures are considered as true nonpure states conceptually distinct from proper mixtures, we show here that proper and improper mixtures can be represented by different density operators in the quaternionic formulation of quantum mechanics, hence they can be distinguished also from a mathematical viewpoint. A simple example related to the quantum theory of measurement is provided.