The de Finetti theorem for test spaces

TL;DR

This paper generalizes the de Finetti theorem to a broad framework called test spaces, unifying classical and quantum cases and exploring the conditions under which such theorems hold or fail.

Contribution

It introduces a de Finetti theorem for exchangeable states on test spaces, encompassing classical and quantum cases within a unified framework.

Findings

Standard classical and quantum de Finetti theorems are special cases.

The test space framework generalizes the conditions for de Finetti theorems.

Discussion of models where the de Finetti theorem does not apply.

Abstract

We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.