Pure and mixed states

TL;DR

This paper reviews the concepts of pure and mixed states across classical and quantum theories, discussing their mathematical foundations, interpretations, and implications, including a quantum-to-classical transition example.

Contribution

It provides a non-specialist accessible overview of pure and mixed states, integrating classical and quantum perspectives with illustrative examples and implications.

Findings

Pure states and mixtures are unified as expectation values on observable algebras.

Illustration of quantum state convergence to classical mixtures as Planck's constant approaches zero.

Discussion on the meaning and implications of pure and mixed states in different theories.

Abstract

We present a review on the notion of pure states and mixtures as mathematical concepts that apply for both classical and quantum physical theories, as well as for any other theory depending on statistical description. Here, states will be presented as expectation values on suitable algebras of observables, in a manner intended for the non-specialist reader; accordingly, basic literature on the subject will be provided. Examples will be exposed together with a discussion on their meanings and implications. An example will be shown where a pure quantum state converges to a classical mixture of particles as Planck's constant tends to zero.