Measuring incompatibility and clustering quantum observables with a quantum switch

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 is a faithful measure of incompatibility for sharp observables and provides a metric on the space of von Neumann measurements. It can be efficiently estimated by letting the measurements act in an indefinite order, using a setup known as the quantum switch. Thanks to these features, the MED can be used in quantum machine learning tasks, such as clustering quantum measurement devices based on their mutual compatibility. 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.