Underlining some limitations of the statistical formalism in quantum mechanics

TL;DR

This paper demonstrates that different ensembles with identical density matrices can have distinct measurable properties, indicating limitations in the statistical formalism of quantum mechanics.

Contribution

It reveals that the density matrix formalism does not fully capture all measurable characteristics of quantum ensembles, highlighting its incompleteness.

Findings

Different ensembles with same density matrix can differ in spin variance

Density matrix formalism may be incomplete in describing quantum ensembles

Discussion on the implications of these limitations

Abstract

We show that two chosen ensembles of spin states, which are differently prepared but are described by the same density matrix in quantum mechanics, do not fully share the same measurable characteristics. One characteristic on which they differ is shown to be the variance of the spin along a given direction. We conclude that the statistical description of an ensemble of states as given by its density matrix, although sufficient in many cases, should be considered incomplete, as it does not fully describe the measurable characteristics of the ensemble. A discussion a posteriori on the problem is provided.