Measurement theory in local quantum physics

TL;DR

This paper develops a foundational measurement theory in local quantum physics by analyzing completely positive instruments, introducing the NEP condition, and exploring their realizability as measuring processes within algebraic quantum field theory.

Contribution

It introduces the normal extension property (NEP) for CP instruments, establishing a correspondence with measuring processes and extending results to a broad class of von Neumann algebras.

Findings

Every CP instrument on an atomic von Neumann algebra has the NEP.

Every CP instrument on an injective von Neumann algebra can be approximated by those with the NEP.

In local quantum physics, most CP instruments can be realized as measuring processes within arbitrary error limits.

Abstract

In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional (AFD) von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts (DHR-DR) theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.