Non-representable quantum measures

TL;DR

This paper investigates the limitations of extending quantum-inspired measures from polymeasures to measures, showing that for dimensions two and higher, such measures cannot generally be globally sigma-additive, correcting previous assumptions.

Contribution

It proves that for dimensions two and above, measures derived from polymeasures cannot generally be globally sigma-additive, correcting a prior misconception.

Findings

Measures for d≥2 cannot be globally sigma-additive

Polymeasures produce grade-d measures via diagonalization

Corrects a previous result claiming sigma-additivity for d=2

Abstract

Grade-$d$ measures on a $\sigma$-algebra $\mathcal{A}\subseteq 2^X$ over a set $X$ are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure $\lambda$ on $(X,\mathcal{A})^d$ produces a grade-$d$ measure as its diagonal $\widetilde{\lambda}(A):=\lambda(A,\cdots,A)$, and we prove that as soon as $d\ge 2$ measures (as opposed to polymeasures) do not suffice: the separate $\sigma$-additivity of a $\lambda$ producing $\mu=\widetilde{\lambda}$ cannot, generally, be amplified to global $\sigma$-additivity. This amends a result in the literature, asserting the contrary in case $d=2$.