Quantum Instruments and Conditioned Observables

TL;DR

This paper explores the theoretical framework of quantum instruments and conditioned observables, unifying these concepts within measurement models and illustrating their relationships through a qubit example, advancing foundational understanding.

Contribution

It introduces a unified framework for quantum instruments and conditioned observables, providing new insights into their relationships and joint probabilities in quantum measurement theory.

Findings

Unified framework for quantum instruments and conditioned observables

Illustrated concepts with a qubit example

Proposed a new definition for joint probabilities

Abstract

Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.