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.

翻译：不相容可观测量的存在是量子力学的基石，也是量子技术中的重要资源。本文提出一种称为互本征空间扰动（MED）的不相容性度量方法，用于量化尖锐可观测量在另一可观测量本征空间上的测量扰动程度。MED为冯·诺依曼测量空间提供了度量标准，并可通过量子开关这一非定序测量装置高效估计，从而实现对任意量子过程非对易性的量化。基于这些特性，MED可应用于量子机器学习任务。我们通过提出一种无监督算法来演示该应用，该算法能对未知冯·诺依曼测量进行聚类，且对噪声具有鲁棒性，可用于识别共享近似相同测量情景的观测者群体。