A Kolmogorov Extension Theorem for POVMs
Roderich Tumulka · 2007
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.