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：不相容观测量是量子力学的基石，也是量子技术中的重要资源。本文引入了一种称为互本征空间扰动（MED）的不相容性度量，该度量量化了在对一个尖锐可观测量进行测量时，对另一个可观测量本征空间产生的扰动程度。MED为冯·诺依曼测量空间提供了度量标准，并可通过让测量过程以不确定顺序作用（利用称为"量子开关"的设置）进行高效估计，该方法还可量化任意量子过程的非对易性。基于这些特性，MED可用于量子机器学习任务。我们通过提供一种对未知冯·诺依曼测量进行聚类的无监督算法验证了这一应用。该算法对噪声具有鲁棒性，可用于识别共享近似相同测量背景的观测者群体。