Measurement Models with Separable Interaction Channels

TL;DR

This paper introduces a separability condition for interaction channels in quantum measurement models, enabling explicit forms of measured observables and analyzing their statistical properties and uncertainty relations.

Contribution

It proposes a separability condition for interaction channels in measurement models, allowing explicit observable forms and detailed analysis of product and conditioned measurement models.

Findings

Separable channels yield explicit forms of measured observables.

Product and conditioned measurement models are studied under separability.

Uncertainty principles for measurement statistics are analyzed.

Abstract

Measurement models (MMs) stand at the highest structural level of quantum measurement theory. MMs can be employed to construct instruments which stand at the next level. An instrument is thought of as an apparatus that is used to measure observables and update states. Observables, which are still at the next level, are used to determine probabilities of quantum events. The main ingredient of an MM is an interaction channel $\nu$ between the system being measured and a probe system. For a general $\nu$, the measured observable $A$ need not have an explicit useful form. In this work we introduce a condition for $\nu$ called separability and in this case $A$ has an explicit form. Under the assumption that $\nu$ is separable, we study product MMs and conditioned MMs. We also consider the statistics of MMs and their uncertainty principle. Various concepts are illustrated using examples of L\"uders and Holevo instruments.