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...

Dynamical quantum phase transitions (DQPTs) extend the concept of phase transitions and thus universality to the non-equilibrium regime. In this letter, we investigate DQPTs in a string of ions simulating interacting transverse-field Ising models. We observe non-equilibrium dynamics induced by a quantum quench and show for strings of up to 10 ions the direct detection of DQPTs by measuring a quantity that becomes non-analytic in time in the thermodynamic limit. Moreover, we provide a link between DQPTs and the dynamics of other relevant quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production.Today, the equilibrium properties of quantum matter are theoretically described with remarkable success. Yet, in recent years pioneering experiments have created novel quantum states beyond this equilibrium paradigm [1,2]. Thanks to this progress, it is now possible to experimentally study exotic phenomena such as many-body localization [3,4], prethermalization [5, 6], particle-antiparticle production in the lattice Schwinger model [7], and light-induced superconductivity [8]. Understanding general properties of such nonequilibrium quantum states provides a significant challenge, calling for new concepts that extend important principles such as universality to the non-equilibrium realm. A general approach towards this major goal is the theory of dynamical quantum phase transitions (DQPTs) [9], which extends the concept of phase transitions and thus universality to the nonequilibrium regime. In this letter, we directly observe the defining real-time non-analyticities at DQPTs in a trappedion quantum simulator for interacting transverse-field Ising models. Moreover, we provide a link between DQPTs and the dynamics of other relevant quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production. Our work advances towards experimentally characterizing nonequilibrium quantum states and their dynamics, by offering general experimental tools that can be applied also to other inherently dynamical phenomena.Statistical mechanics and thermodynamics provide us with an excellent understanding of equilibrium quantum manybody systems. A key concept in this framework is the canonical partition function Z(T ) = Tr(e −H/k B T ), with T the temperature, k B the Boltzmann constant, and H the system Hamiltonian. The partition function encodes thermodynamics via the free-energy density f = −(k B T/N) log [Z(T )], where N de- * Present address:ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Sydney, NSW, 2006, Australia notes the number of degrees of freedom. A phase transition, i.e., a sudden change of macroscopic behaviour, is associated with a non-analytical behaviour of f as a function of temperature or another control parameter g such as an external magnetic field. Quantum phase transitions (QPTs) [10] occur when T is kept at absolute zero, where the system's groundstate properties undergo a non-analyt...