A Kolmogorov Extension Theorem for POVMs

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.