# Dual Quantum Instruments and Sub-observables

## TL;DR

This paper introduces dual quantum instruments and sub-observables, exploring their properties, relationships, and extensions, with applications to various quantum measurement models and future research directions.

## Contribution

It defines dual instruments and sub-observables, characterizes their effect algebras, and studies their convexity, sequential products, and extensions, advancing the mathematical framework of quantum measurements.

## Key findings

- Dual instruments measure a unique observable but determine many sub-observables.
- Characterization of sub-observable effect algebras and their convexity.
- Discussion of sequential products and examples including L"uders, Holero, and constant state instruments.

## Abstract

We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss L\"uders, Holero and constant state instruments. Various conjectures for future research are presented.