Non-representable quantum measures

• URL: http://arxiv.org/abs/2508.14326v1
• Date: Wed, 20 Aug 2025 00:47:24 GMT
• Title: Non-representable quantum measures
• Authors: Alexandru Chirvasitu,
• Abstract summary: Grade-$d$ measures on a $sigma$-algebra $mathcalAsubseteq 2X$ over a set $X$ are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions.<n>Every signed polymeasure $lambda$ on $(X,mathcalA)d$ produces a grade-$d$ measure as its diagonal $widetildelambda(A):=lambda(A,cdots,A)$.
• Score: 55.2480439325792
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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