Abstract
We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G n on \({\mathbb{R}}^n\) satisfies the consistency (or projectivity) condition \(G_{n+1}(A\times {\mathbb{R}}) = G_n(A)\) then there is a POVM G on the space \({\mathbb{R}}^{\mathbb{N}}\) of infinite sequences that has G n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory.
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The main proof in this article was first formulated in my habilitation thesis [6].
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Tumulka, R. A Kolmogorov Extension Theorem for POVMs. Lett Math Phys 84, 41–46 (2008). https://doi.org/10.1007/s11005-008-0229-8
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DOI: https://doi.org/10.1007/s11005-008-0229-8