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...

We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimension. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in exisiting AMO quantum simulators, and used to measure for instance area law scaling of entanglement in two-dimensional spin models or the entanglement growth in many-body localized systems.Atomic physics provides us with the realization of engineered quantum many-body lattice models. This includes Hubbard models for bosonic and fermionic cold atoms in optical lattices [1], and spin models with Rydberg atoms [2] and chains of trapped ions [3]. Among the noticeable recent experimental advances are quantum control, and single shot measurements in lattice systems of atoms [4][5][6][7][8][9][10][11] and ions [12,13] achieving single site resolution, as illustrated for atoms in optical lattices by the quantum gas microscope [14]. This provides us not only with a unique atomic toolbox to prepare equilibrium and non-equilibrium states of quantum matter, but also with the opportunity to access in experiments novel classes of observables, beyond the familiar low order correlation functions. An outstanding example is the measurement of Rényi entropies, defined as S (n) (ρ A ) = 1 1−n log Tr(ρ n A ) (n > 1) with ρ A = Tr S\A [ρ] the reduced density matrix of a subsystem A ⊂ S of a many-body system S, which gives us a unique signature of entanglement properties in many-body phases and dynamics [15], and is also of interest in the ongoing discussion on 'quantum supremacy' [16][17][18][19][20].Below we will describe a protocol for measuring Rényi entropies S (n) (ρ A ) based on random measurements realized as random unitary operators applied to ρ A and subsequent measurements of a fixed observable [21]. In our approach the required random unitaries are implemented using the same AMO toolbox which underlies the preparation of quantum phases and dynamics (c.f. Fig. 1). This enables a physical implementation of the protocol, applicable to generic Hubbard and spin models and in arbitrary dimension. We emphasize that in contrast to recent protocols to measure n-th order Rényi entropies, which requires preparation of n identical copies [22][23][24][25], a random measurement protocol requires only a single quantum system [21], and thus can be implemented directly with existing AMO and solid state platforms [26,27]. A central aspect in any measurement scheme for Rényi

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