A Kolmogorov Extension Theorem for POVMs

TL;DR

This paper establishes a Kolmogorov extension theorem analog for POVMs, enabling the construction of a consistent infinite-dimensional POVM from a sequence of compatible finite-dimensional POVMs, with applications in quantum theory.

Contribution

It introduces a new extension theorem for POVMs, generalizing classical probability results to quantum measurement frameworks.

Findings

Proves a Kolmogorov extension theorem for POVMs.

Constructs a POVM on infinite sequences from finite-dimensional POVMs.

Provides an application in quantum theory.

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.