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.Engineered quantum systems, consisting of tens of individually-controllable interacting quantum particles, are currently being developed using a number of different physical platforms; including atoms in optical arrays (1-3), ions in radio-frequency traps (4, 5), and superconducting circuits (6-9). These systems offer the possibility of generating and 1 arXiv:1806.05747v2 [quant-ph] 14 Jan 2019 probing complex quantum states and dynamics particle by particle -finding application in the near-term as quantum simulators, and in the longer-term as quantum computers. As these systems are developed, new protocols are required to characterize them -to verify that they are performing as desired and to measure quantum phenomena of interest.A key property to measure in engineered quantum systems is entanglement. For example, in order for quantum simulators and computers to provide an advantage over their classical analogues, they must generate large amounts of entanglement between their parts (10). Furthermore, when using these devices to tackle open questions in physics, the dynamics of entanglement provides signatures of the phenomena of interest, such as thermalization (11) and many-body localization (12,13).Entanglement can be probed by measuring entanglement entropies. In particular, consider the second-order Rényi entropywith ρ A the reduced density matrix for a part A of the total system described by ρ. If the entropy of part A is greater than the entropy of the total system; i.e S (2) (ρ A ) > S (2) (ρ), bipartite entanglement exists between A and the rest of the system (14). Thus, a measurement of the entropy of the whole system, as well as of its subsystems, provides information about the entanglement contained within the system. Additionally, a measurement of the entropy of the total state ρ provides the opportunity to verify the overall coherence of the system, as for pure quantum states S (2) (ρ) = 0.Recently, a protocol to directly measure the second-order Rényi entropy, S (2) , has been demonstrated, requiring collective measurements to be made on two identical copies ρ of a quantum system (15)(16)(17)(18). In (17), that protocol was used to study entanglement growth and thermali...

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We describe a protocol for cross-platform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr [ρ1ρ2] and the purities Tr ρ 2 1,2 , and thus a (mixed-state) fidelity, of two quantum states ρ1 and ρ2 prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proof-of-principle, we present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator.