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arXiv · 2508.14326

Non-representable quantum measures

Alexandru Chirvasitu
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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 $d=2$.

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Alexandru Chirvasitu. 2025-08-20. Non-representable quantum measures. https://arxiv.org/abs/2508.14326

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