Analysing quantum systems with randomised measurements

Measurements with randomly chosen settings determine many important properties of quantum states without the need for a shared reference frame or calibration. They naturally emerge in the context of quantum communication and quantum computing when dealing with noisy environments, and allow the estimation of properties of complex quantum systems in an easy and efficient manner. In this review, we present the advancements made in utilising randomised measurements in various scenarios of quantum information science. We describe how to detect and characterise different forms of entanglement, including genuine multipartite entanglement and bound entanglement. Bell inequalities are discussed to be typically violated even with randomised measurements, especially for a growing number of particles and settings. Furthermore, we also present an overview on the estimation of non-linear functions of quantum states and shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.