  Entanglement is the key feature of many-body quantum systems, and the
development of new tools to probe it in the laboratory is an outstanding
challenge. Measuring the entropy of different partitions of a quantum system
provides a way to probe its entanglement structure. Here, we present and
experimentally demonstrate a new protocol for measuring entropy, based on
statistical correlations between randomized measurements. Our experiments,
carried out with a trapped-ion quantum simulator, prove the overall coherent
character of the system dynamics and reveal the growth of entanglement between
its parts - both in the absence and presence of disorder. Our protocol
represents a universal tool for probing and characterizing engineered quantum
systems in the laboratory, applicable to arbitrary quantum states of up to
several tens of qubits.
