  We develop a general theoretical framework for measurement protocols
employing statistical correlations of randomized measurements. We focus on
locally randomized measurements implemented with local random unitaries in
quantum lattice models. In particular, we discuss the theoretical details
underlying the recent measurement of the second R\'{e}nyi entropy of highly
mixed quantum states consisting of up to $10$ qubits in a trapped-ion quantum
simulator [Brydges et al., Science 364, 260 (2019)]. We generalize the protocol
to access the overlap of quantum states, prepared sequentially in an
experiment. Furthermore, we discuss proposals for quantum state tomography
based on randomized measurements within our framework and the respective
scaling of statistical errors with system size.
