Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality

TL;DR

This paper proposes interpreting density matrices as quasi-moment matrices within a new formulation of quantum theory, linking quantum phenomena with classical probability concepts and explaining the emergence of classical reality from quantum systems.

Contribution

It introduces a novel formulation of quantum theory using quasi-expectation operators, unifying classical and quantum representations and explaining classical emergence from quantum systems.

Findings

Density matrices can be interpreted as quasi-moment matrices.

Quantum indistinguishability relates to classical exchangeability.

Classical reality emerges as the number of identical particles increases.

Abstract

We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs). This formulation provides a direct interpretation of density matrices as quasi-moment matrices. Using QEOs, we will provide a series of representation theorems, a' la de Finetti, relating a classical probability mass function (satisfying certain symmetries) to a quasi-expectation operator. We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. Although particles indistinguishability is considered a truly "weird" quantum phenomenon, it is not special. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT. Using this connection, we will rederive the first and second quantisation in QT for bosons through the classical statistical concept of exchangeable random variables. Using this approach, we will show how classical reality emerges in QT as the number of identical bosons increases (similar to what happens for finitely exchangeable sequences of rolls of a classical dice).