Analysing quantum systems with randomised measurements

Randomised measurements provide a way of determining physical quantities without the need for a shared reference frame nor calibration of measurement devices. Therefore, they naturally emerge in situations such as benchmarking of quantum properties in the context of quantum communication and computation where it is difficult to keep local reference frames aligned. In this review, we present the advancements made in utilising such measurements in various quantum information problems focusing on quantum entanglement and Bell inequalities. We describe how to detect and characterise various 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. Additionally, we provide an overview of estimating other relevant nonlinear functions of a quantum state or performing shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.