Measuring incompatibility and clustering quantum observables with a quantum switch

TL;DR

This paper introduces the mutual eigenspace disturbance (MED), a new measure of quantum observable incompatibility, and demonstrates its estimation via a quantum switch to enable clustering of quantum measurements.

Contribution

The paper presents the MED as a novel incompatibility measure and shows how to estimate it using a quantum switch, enabling quantum measurement clustering.

Findings

MED quantifies measurement disturbance and incompatibility.

Quantum switch allows efficient estimation of MED.

Clustering algorithm groups similar quantum measurements.

Abstract

The existence of incompatible observables is a cornerstone of quantum mechanics and a valuable resource in quantum technologies. Here we introduce a measure of incompatibility, called the mutual eigenspace disturbance (MED), which quantifies the amount of disturbance induced by the measurement of a sharp observable on the eigenspaces of another. The MED provides a metric on the space of von Neumann measurements, and can be efficiently estimated by letting the measurement processes act in an indefinite order, using a setup known as the quantum switch, which also allows one to quantify the noncommutativity of arbitrary quantum processes. Thanks to these features, the MED can be used in quantum machine learning tasks. We demonstrate this application by providing an unsupervised algorithm that clusters unknown von Neumann measurements. Our algorithm is robust to noise can be used to identify groups of observers that share approximately the same measurement context.