# Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states

## TL;DR

This paper introduces a comprehensive framework for using statistical correlations of randomized measurements to probe entanglement and quantum states in many-body systems, with applications demonstrated in quantum simulators.

## Contribution

It develops a general theoretical approach for measurement protocols using randomized measurements, extending previous work to access state overlaps and tomography.

## Key findings

- Successfully measured second Rényi entropy in a 10-qubit trapped-ion system.
- Generalized protocols to access quantum state overlaps.
- Discussed scalable quantum state tomography methods.

## Abstract

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.