Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states. (arXiv:1812.02624v1 [quant-ph])

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., arXiv:1806.05747]. 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.