Simultaneous Measurement of Two Quantum Observables: Compatibility, Broadcasting, and In-between

TL;DR

This paper explores the hierarchy of quantum observable relations, including compatibility and broadcastability, providing a detailed classification and characterization for qubit observables to understand their measurement limitations.

Contribution

It introduces a hierarchy of four symmetric relations stronger than compatibility and characterizes them for qubit observables, clarifying their measurement requirements.

Findings

Hierarchy of five relations including compatibility and broadcastability.

Complete characterization of stronger relations for qubit observables.

Clarification of measurement device requirements for each relation.

Abstract

One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance and nondisturbance. It is proven that the five relations form a hierarchy, and their differences in terms of the required devices needed in a simultaneous measurement is explained. All the four relations stronger than compatibility are completely characterized in the case of qubit observables.