An entropic look into entanglement Quantum systems are predicted to be better at information processing than their classical counterparts, and quantum entanglement is key to this superior performance. But how does one gauge the degree of entanglement in a system? Brydges et al. monitored the build-up of the so-called Rényi entropy in a chain of up to 10 trapped calcium ions, each of which encoded a qubit. As the system evolved, interactions caused entanglement between the chain and the rest of the system to grow, which was reflected in the growth of the Rényi entropy. Science, this issue p. 260

Abstract Entanglement is a key feature of many-body quantum systems. 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 protocol for measuring the second-order Rényi entropy based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator with partition sizes of up to 10 qubits, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts, in both the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, which is applicable to arbitrary quantum states of up to several tens of qubits.

Engineered quantum systems that consist 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 probing complex quantum states and dynamics particle by particle and are finding application in the near term as quantum simulators and in the longer term as quantum computers. As these systems are developed, more and more sophisticated protocols are required to characterize them—i.e., 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, for quantum simulators and computers to provide an advantage over their classical analogs, they must generate large amounts of entanglement between their parts (10). Furthermore, entanglement provides signatures of a wide range of phenomena, including quantum criticality and topological phases (11) as well as thermalization dynamics (12) and many-body localization (13, 14). In addition, entanglement underpins the working mechanism of widely used numerical methods based on tensor network states (11).

Entanglement can be probed by measuring entanglement entropies. In particular, consider the second-order Rényi entropy S ( 2 ) ( ρ A ) = − log 2 Tr ( ρ A 2 ) (1)with ρ A as 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 (15). Thus, measuring the entropy of the whole system and that of its subsystems provides information about the entanglement contained in the system. Additionally, a measurement of the entropy of the total state ρ provides a test of the overall purity 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 (16–19). In (18), that protocol was used to study entanglement growth and thermalization in a six-site Bose-Hubbard system, realized with atoms in an optical lattice.

Here, we introduce and experimentally demonstrate a different protocol to measure the second-order Rényi entropy S ( 2 ) , which is based on and extends the proposals in (20–23). Key strengths of the protocol are that it requires preparation of only a single copy of the quantum system at a time and can be implemented on any physical platform with single-particle readout and control. In contrast to recently developed, efficient tomographic methods (24, 25) to characterize weakly entangled states, our approach imposes no a priori assumption on the structure of the quantum state. Instead, it provides direct access to properties of the density matrix that are invariant under local unitary transformations, such as S ( 2 ) , without the need for prior tomographic reconstruction. S ( 2 ) can therefore be estimated with a significantly lower number of measurements than is necessary for quantum state tomography (see last paragraph). In our experiments, we used the protocol to measure the dynamical evolution of entanglement entropy of up to 10-qubit partitions of a trapped-ion quantum simulator.

The key insight of the protocol is that information about the second-order Rényi entropies of a system is contained in statistical correlations between the outcomes of measurements performed in random bases. Specifically, for a system of N qubits, the approach (21) is to apply a product of single-qubit unitaries U = u 1 ⊗ ... ⊗ u N , where each unitary u i is drawn independently from the circular unitary ensemble (CUE) (26), and then to measure the qubits in a fixed (logical) basis. For each U, repeated measurements are made to obtain statistics, and the entire process is repeated for many different randomly drawn instances of U. The second-order Rényi entropy, S ( 2 ) , of the density matrix ρ A for an arbitrary partition A = { i [ 1 ] , ... , i [ N A ] } of N A ≤ N qubits is then obtained from S ( 2 ) ( ρ A ) = − log 2 X ¯ , with X = 2 N A ∑ s A , s ′ A A ( − 2 ) − D [ s A , s ' A ] P ( s A ) P ( s ′ A ) (2)

In Eq. 2, the bar denotes the ensemble average of (cross-) correlations of excitation probabilities P ( s A ) = 〈 s A | U A ρ A U A † | s A 〉 ; s A are the logical basis states of partition A, U A = U | A is the restriction of U to A, D [ s A , s ' A ] is the Hamming distance between s A and s ' A , and † represents the Hermitian conjugate. X ¯ is equal to the purity Tr ( ρ A 2 ) of the density matrix ρ A . Equation 2 represents an explicit formula, proven in the supplementary text (27), to reconstruct the second-order Rényi entropy of the subsystem of interest directly from statistical correlations between randomized measurements. As a result, compared with the recursive scheme presented in (21), an exponential overhead in the classical postprocessing is avoided.

For the partition of a single qubit, N A = 1, the Bloch sphere provides a simple graphical representation to clarify the relation between the purities and the distribution of excitation probabilities (Fig. 1A). For a pure state, Tr ( ρ A 2 ) = 1 , the quantum state can be represented as a unit Bloch vector on the sphere, with random rotations leading to a uniform distribution of probabilities covering the full range [0, 1]. For a mixed state, Tr ( ρ A 2 ) < 1 , the length of the Bloch vector is less than 1, and the probabilities take values in a reduced interval. Generalizing to the multiqubit scenario, the purities are directly inferred from the mean of the statistical distribution of a weighted sum of cross-correlations by using Eq. 2. Examples of cross-correlations that were measured for different partition sizes of the trapped-ion system are shown in Fig. 1B, together with the estimated purities.

Fig. 1 Measuring second-order Rényi entropies through randomized measurements. (A) Single-qubit Bloch sphere. The purity is directly related to the width of the distribution of measurement outcomes after applying random rotations u i . Initial pure state (blue) and mixed state (red) cases are shown. See text. (B) Generalization to multiple qubits: Measuring N A -qubit (up to 10) partitions of a 20-qubit string, as shown (top). Repeated measurements (N M = 150 and N U = 500) were made to obtain statistics; see text. Experimental data (bottom): Histograms of the weighted sum X of cross-correlations (as defined in Eq. 2), with mean values corresponding to the purities (dashed lines). Results are shown for two different times during evolution under H XY (Eq. 3), starting from a highly pure, separable state (blue) and evolving into a high-entropy state (red).

Our experiments were implemented by using strings of up to 20 trapped 40Ca+ ions, each of which encodes a qubit that can be individually manipulated by spatially focused, coherent laser pulses. When dressed with suitably tailored laser fields, the ions are subject to a quantum evolution that is equivalent to a model of spins interacting through a long-range XY model (28) in the presence of a transverse field H XY = ℏ ∑ i < j J i j ( σ i + σ j − + σ i − σ j + ) + ℏ B ∑ j σ j z (3)Here, ℏ is Planck’s constant divided by 2π, σ i β ( β = x , y , z ) are the spin-½ Pauli operators, σ i + ( σ i − ) the spin-raising (lowering) operators acting on spin i, and J i j ≈ J 0 / | i − j | α the coupling matrix with an approximate power-law decay and 0 < α < 3 . For further experimental details, see (27, 29). Optionally, a locally disordered potential could be added (30, 31), realizing the Hamiltonian H = H XY + H D , with H D = ℏ ∑ j Δ j σ j z and Δ j the magnitude of disorder applied to ion j. For entropy measurements, the following experimental protocol was used throughout: The system was initially prepared in the Néel ordered product state ρ 0 = | ψ 〉 〈 ψ | with | ψ 〉 = | ↓ ↑ ↓ ... ↑ 〉 . This state was subsequently time-evolved under H XY (or H) into the state ρ ( t ) . The coherent interactions arising from this time evolution generated varying types of entanglement in the system. Subsequently, randomized measurements on ρ ( t ) were performed through individual rotations of each qubit by a random unitary (u i ), sampled from the CUE (26), followed by a state measurement in the z basis. Each u i can be decomposed into three rotations R z ( θ 3 ) R y ( θ 2 ) R z ( θ 1 ) , and two random unitaries were concatenated to ensure that drawing of the u i was stable against small drifts of physical parameters controlling the rotation angles θ i (27). Finally, spatially resolved fluorescence measurements realized a projective measurement in the logical z basis. To measure the entropy of a quantum state, N U sets of single-qubit random unitaries, U = u 1 ⊗ … ⊗ u N , were applied. For each set of applied unitaries, U, the measurement was repeated N M times.

In the first experiment, a 10-qubit state ρ 0 was prepared and subsequently time-evolved under H XY (Eq. 3), without disorder, for t = 0 , … , 5 ms . Figure 2, A and B, respectively show measured purities and entropies of all connected partitions that include qubit 1 during this quench. The overall purity (and thus entropy) remained at a constant value of Tr [ ρ 2 ] = 0.74 ± 0.07 , within error, throughout the time evolution, implying that the time evolution was approximately unitary. The initial state’s reconstructed purity is in agreement with control experiments, which show a purity loss of 0.08 caused by imperfect state preparation and an underestimation of the purity by ~0.17 caused by decoherence during the random spin rotations (27). At short times, the single-spin subsystem became quickly entangled with the rest of the system, seen as a rapid decrease (increase) of the single-spin purity (entropy) (Fig. 2, A and B), until the reduced state became completely mixed. At longer times, the purity (entropy) of larger subsystems continued to decrease (increase), as they became entangled with the rest. The dotted curves represent numerical simulations for the experimental parameters, including decoherence, during state initialization, evolution, and measurement (27). Although Fig. 2, A and B, correspond to a specific set of connected partitions A, the data give access to the purities for all partitions A of the system, as shown in Fig. 2C for a specific time t = 5 ms. Because the second-order Rényi entropy of every subsystem is, within three standard deviations, larger than for the total system, this demonstrates entanglement between all 29 – 1 = 511 bipartitions of the 10-qubit system.

Fig. 2 Purity and second-order Rényi entropies of a 10-qubit system. (A) Measured purity and (B) second-order Rényi entropy of a Néel state, time-evolved under H XY (J 0 = 420 s−1, α = 1.24), for connected partitions [ 1 → i ] . Dotted curves are purities derived from a numerical simulation; see supplementary materials (27). Maximally mixed states with minimal purity fall on the boundary of the shaded area. (C) Second-order Rényi entropy, S ( 2 ) ( ρ A ) , of all 210 − 1 = 1023 partitions at t = 5 ms, with N A denoting the number of ions in a partition A. For all data points, N M = 150 and N U = 500. Error bars, which increase with subsystem size (27), are standard errors of the mean X ¯ . Lines in (C) are drawn at three standard errors above the full system’s entropy (black, dashed) and three standard errors below the minimal subsystem’s entropy (blue, solid).

Next, a 20-qubit experiment was performed, in which the entropy growth of the central part of the chain was measured during time evolution under H XY , for partitions of up to 10 qubits. Our observations (Fig. 3) are consistent with the formation of highly entangled states. The entropy increases rapidly over the time evolution of 10 ms, with the reduced density matrices of up to seven qubits becoming nearly fully mixed. The experimental data agree very well with numerical simulations (dotted curves) obtained with a matrix-product state algorithm (32), which includes the (weak) effect of decoherence using quantum trajectories (33). The measurement highlights the ability of our protocol to access the entropy of highly mixed states, despite larger statistical errors compared with pure states (27).

Fig. 3 Second-order Rényi entropy of 1- to 10-qubit partitions of a 20-qubit system. The initial low-entropy Néel state evolves under H XY (J 0 = 370 s−1, α = 1.01) within 10 ms into a state with high-entropy partitions, corresponding to nearly fully mixed subsystems. For the data taken at 6 ms (10 ms) of time evolution, the two (three) data points corresponding to highly mixed states are not shown, because they have large statistical error bars. For details regarding numerical simulations (dotted curves) and error bars, see (27).

Monitoring the entropy growth of arbitrary yet highly entangled states during their time evolution constitutes a universal tool for studying dynamical properties of quantum many-body systems, in connection with the concept of quantum thermalization (12). In this context, a slow entropy growth can be used to signify localization in generic many-body quantum systems (14). Generically, in interacting quantum systems without disorder, a ballistic (linear) entropy growth is predicted after a quantum quench (12). Such growth is assumed to persist until saturation is reached, signaling thermalization of the system at late times. On the contrary, in the presence of (strong) disorder and sufficiently short-ranged interactions, the existence of the many-body localized (MBL) phase (13) is predicted in one-dimensional systems (34). This phase is characterized by the absence of thermalization, the system’s remembrance on the initial state (35) at late times, and, in particular, a logarithmic entropy growth (36, 37), which constitutes the distinguishing feature between an MBL state and a noninteracting Anderson insulator. Experiments probing this entropy growth have been realized with superconducting qubits by using tomography (8) and ultracold atoms based on full-counting statistics of particle numbers (38). The persistence and stability of localization in long-range interacting systems have also been explored, both theoretically (14, 34, 39) and experimentally (30). The measurement of a long-time entropy growth rate is beyond the present capabilities of our trapped-ion quantum simulator, owing to its limited coherence time; however, we were able to observe the effects of local, random disorder on the entropy growth rate at early times.

Figure 4A displays the measured evolution of the second-order Rényi entropy at half partition as a function of time, both in the absence and in the presence of local random disorder. Without disorder, a rapid, linear growth of entropy is observed, in agreement with theoretical simulations including the mentioned sources of decoherence (solid lines). To investigate the influence of disorder, the initial Néel state was quenched with H = H XY + H D , where the static, random disorder strength Δ j was drawn uniformly from [ − 3 J 0 , 3 J 0 ] . To efficiently access directly disorder-averaged quantities, our protocol offers the possibility to reduce the number of random unitaries that must be applied per disorder pattern and instead average in addition over different disorder patterns (27). Hence, only 10 random unitaries per disorder pattern (N M = 150 measurements per unitary) and 35 randomly drawn disorder patterns were used to obtain an accurate estimate of the disorder-averaged purity Tr [ ρ A 2 ] ˜ ( ... ˜ denotes the disorder average) and subsequently the second-order Rényi entropy S ( 2 ) ( ρ A ) ˜ ≈ − log 2 Tr [ ρ A 2 ] ˜ (27). The measured, disorder-averaged entropy growth clearly demonstrates how disorder reduces the growth of entanglement. After an initial rapid evolution, a considerable slowing of the dynamics is observed, with a small but nonvanishing growth rate at later times, a behavior consistent with the scenario of MBL. The system retains memory of the initial Néel state during the dynamics, which is manifest in the measured time evolution of the local magnetization (fig. S5) (27).

Fig. 4 Spread of quantum correlations under H XY with and without disorder. The Hamiltonian parameters are J 0 = 420 s−1, α = 1.24. (A) Half-chain entropy growth versus time without disorder (red data points) and with disorder, drawn uniformly from [ − 3 J 0 , 3 J 0 ] (blue data points). Numerical simulations based on unitary dynamics (dotted curves) including known sources of decoherence (full lines) are in agreement with the measured second-order Rényi entropies [see supplementary materials (27)]. (B) Second-order RMI of selected subsystems versus time in the presence of disorder (see Eq. 4). The decrease of I ( 2 ) with distance between subsystems is a manifestation of the inhibition of correlation spreading by local disorder. For longer time scales, decoherence leads to a slow increase in the entropy of the total system [ S ( 2 ) ( ρ ) ≈ 0.9 for t = 10 ms for the full system (27)]. Consequently, there is an additional contribution to the slow entropy growth of the system from this decoherence, compared with the case of purely unitary dynamics. Error bars are the standard error of the mean, calculated with jackknife resampling of the applied random unitaries (40).

Finally, Fig. 4B shows the evolution of the second-order Rényi mutual information (RMI), defined as I ( 2 ) ( ρ A : ρ B ) = S ( 2 ) ( ρ A ) + S ( 2 ) ( ρ B ) − S ( 2 ) ( ρ A B ) (4)In the presence of disorder, the RMI saturates quickly to approximately constant values, which decrease with increasing distance between the two partitions A and B. This indicates a spatial decay of correlations in the system, a characteristic feature of localization caused by the presence of disorder; this conclusion is supported by a numerical comparison of the RMI to the von Neumann mutual information, showing that they behave in qualitatively the same way (27).

In our experiments, we studied the entropy of partitions of up to 10 qubits because technical restrictions currently limit our experimental repetition rate. Straightforward technical improvements should allow the entropy of 20-qubit systems to be measurable. Numerical simulations (27) indicate that the total number of measurements required to access the purity within a statistical error of 0.12 is, for a pure product state of N A qubits, given by 2 7.7 ± 0.3 + ( 0.8 ± 0.1 ) N A . The number of measurements required to obtain the purity of entangled pure states can be significantly lower (27). Purity measurements of systems containing tens of qubits are likely also in reach in experiments with high quantum state–generation rates, such as state-of-the-art superconducting qubit setups. The number of measurements could be further decreased by replacing the local random operations by global random unitaries acting on the entire Hilbert space of a subsystem of interest, by means of random quenches (21, 22), at the expense of obtaining access to the purity of a single partition only.

Supplementary Materials science.sciencemag.org/content/364/6437/260/suppl/DC1 Supplementary Text Figs. S1 to S8 References (43–46)

http://www.sciencemag.org/about/science-licenses-journal-article-reuse This is an article distributed under the terms of the Science Journals Default License.

References and Notes ↵ I. Bloch , J. Dalibard , S. Nascimbène , Quantum simulations with ultracold quantum gases . Nat. Phys. 8 , 267 – 276 ( ). doi: 10.1038/nphys2259 OpenUrl CrossRef A. Browaeys , D. Barredo , T. Lahaye , Experimental investigations of dipole–dipole interactions between a few Rydberg atoms . J. Phys. B 49 , 152001 ( ). doi: 10.1088/0953-4075/49/15/152001 OpenUrl CrossRef ↵ M. Saffman , Quantum computing with atomic qubits and Rydberg interactions: Progress and challenges . J. Phys. B 49 , 202001 ( ). doi: 10.1088/0953-4075/49/20/202001 OpenUrl CrossRef ↵ J. Zhang , G. Pagano , P. W. Hess , A. Kyprianidis , P. Becker , H. Kaplan , A. V. Gorshkov , Z.-X. Gong , C. Monroe , Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator . Nature 551 , 601 – 604 ( ). doi: 10.1038/nature24654 pmid: 29189781 OpenUrl CrossRef PubMed ↵ N. Friis , O. Marty , C. Maier , C. Hempel , M. Holzäpfel , P. Jurcevic , M. B. Plenio , M. Huber , C. Roos , R. Blatt , B. Lanyon , Observation of Entangled States of a Fully Controlled 20-Qubit System . Phys. Rev. X 8 , 021012 ( ). doi: 10.1103/PhysRevX.8.021012 OpenUrl CrossRef ↵ M. Fitzpatrick , N. M. Sundaresan , A. C. Y. Li , J. Koch , A. Houck , Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice . Phys. Rev. X 7 , 011016 ( ). doi: 10.1103/PhysRevX.7.011016 OpenUrl CrossRef J. M. Gambetta , J. M. Chow , M. Steffen , Building logical qubits in a superconducting quantum computing system . npj Quantum Inf. 3 , 2 ( ). doi: 10.1038/s41534-016-0004-0 OpenUrl CrossRef ↵ K. Xu , J.-J. Chen , Y. Zeng , Y.-R. Zhang , C. Song , W. Liu , Q. Guo , P. Zhang , D. Xu , H. Deng , K. Huang , H. Wang , X. Zhu , D. Zheng , H. Fan , Emulating Many-Body Localization with a Superconducting Quantum Processor . Phys. Rev. Lett. 120 , 050507 ( ). doi: 10.1103/PhysRevLett.120.050507 pmid: 29481152 OpenUrl CrossRef PubMed ↵ C. Neill , P. Roushan , K. Kechedzhi , S. Boixo , S. V. Isakov , V. Smelyanskiy , A. Megrant , B. Chiaro , A. Dunsworth , K. Arya , R. Barends , B. Burkett , Y. Chen , Z. Chen , A. Fowler , B. Foxen , M. Giustina , R. Graff , E. Jeffrey , T. Huang , J. Kelly , P. Klimov , E. Lucero , J. Mutus , M. Neeley , C. Quintana , D. Sank , A. Vainsencher , J. Wenner , T. C. White , H. Neven , J. M. Martinis , A blueprint for demonstrating quantum supremacy with superconducting qubits . Science 360 , 195 – 199 ( ). doi: 10.1126/science.aao4309 pmid: 29650670 OpenUrl ↵ G. Vidal , Efficient classical simulation of slightly entangled quantum computations . Phys. Rev. Lett. 91 , 147902 ( ). doi: 10.1103/PhysRevLett.91.147902 pmid: 14611555 OpenUrl CrossRef PubMed ↵ J. Eisert , M. Cramer , M. B. Plenio , Colloquium: Area laws for the entanglement entropy . Rev. Mod. Phys. 82 , 277 – 306 ( ). doi: 10.1103/RevModPhys.82.277 OpenUrl CrossRef Web of Science ↵ P. Calabrese , J. Cardy , Evolution of entanglement entropy in one-dimensional systems . J. Stat. Mech. 2005 , P04010 ( ). doi: 10.1088/1742-5468/2005/04/P04010 OpenUrl CrossRef ↵ D. M. Basko , I. L. Aleiner , B. L. Altshuler , Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states . Ann. Phys. 321 , 1126 – 1205 ( ). doi: 10.1016/j.aop.2005.11.014 OpenUrl CrossRef ↵ D. A. Abanin, E. Altman, I. Bloch, M. Serbyn, Ergodicity, entanglement and many-body localization. arXiv:1804.11065v1 [cond-mat.dis-nn] (30 April 2018). ↵ R. Horodecki , P. Horodecki , M. Horodecki , K. Horodecki , Quantum entanglement . Rev. Mod. Phys. 81 , 865 – 942 ( ). doi: 10.1103/RevModPhys.81.865 OpenUrl CrossRef ↵ A. K. Ekert , C. M. Alves , D. K. L. Oi , M. Horodecki , P. Horodecki , L. C. Kwek , Direct estimations of linear and nonlinear functionals of a quantum state . Phys. Rev. Lett. 88 , 217901 ( ). doi: 10.1103/PhysRevLett.88.217901 pmid: 12059503 OpenUrl CrossRef PubMed R. Islam , R. Ma , P. M. Preiss , M. E. Tai , A. Lukin , M. Rispoli , M. Greiner , Measuring entanglement entropy in a quantum many-body system . Nature 528 , 77 – 83 ( ). doi: 10.1038/nature15750 pmid: 26632587 OpenUrl CrossRef PubMed ↵ A. M. Kaufman , M. E. Tai , A. Lukin , M. Rispoli , R. Schittko , P. M. Preiss , M. Greiner , Quantum thermalization through entanglement in an isolated many-body system . Science 353 , 794 – 800 ( ). doi: 10.1126/science.aaf6725 pmid: 27540168 OpenUrl ↵ N. M. Linke , S. Johri , C. Figgatt , K. A. Landsman , A. Y. Matsuura , C. Monroe , Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer . Phys. Rev. A 98 , 052334 ( ). doi: 10.1103/PhysRevA.98.052334 OpenUrl CrossRef ↵ S. J. van Enk , C. W. J. Beenakker , Measuring Trρn on single copies of ρ using random measurements . Phys. Rev. Lett. 108 , 110503 ( ). doi: 10.1103/PhysRevLett.108.110503 pmid: 22540446 OpenUrl CrossRef PubMed ↵ A. Elben , B. Vermersch , M. Dalmonte , J. I. Cirac , P. Zoller , Rényi Entropies from Random Quenches in Atomic Hubbard and Spin Models . Phys. Rev. Lett. 120 , 050406 ( ). doi: 10.1103/PhysRevLett.120.050406 pmid: 29481179 OpenUrl CrossRef PubMed ↵ B. Vermersch , A. Elben , M. Dalmonte , J. I. Cirac , P. Zoller , Unitary n-designs via random quenches in atomic Hubbard and spin models: Application to the measurement of Rényi entropies . Phys. Rev. A 97 , 023604 ( ). doi: 10.1103/PhysRevA.97.023604 OpenUrl CrossRef ↵ A. Elben, B. Vermersch, C. F. Roos, P. Zoller, Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states. arXiv:1812.02624 [quant-ph] (6 Dec 2018). ↵ B. P. Lanyon , C. Maier , M. Holzäpfel , T. Baumgratz , C. Hempel , P. Jurcevic , I. Dhand , A. S. Buyskikh , A. J. Daley , M. Cramer , M. B. Plenio , R. Blatt , C. F. Roos , Efficient tomography of a quantum many-body system . Nat. Phys. 13 , 1158 – 1162 ( ). doi: 10.1038/nphys4244 OpenUrl CrossRef ↵ G. Torlai , G. Mazzola , J. Carrasquilla , M. Troyer , R. Melko , G. Carleo , Neural-network quantum state tomography . Nat. Phys. 14 , 447 – 450 ( ). doi: 10.1038/s41567-018-0048-5 OpenUrl CrossRef ↵ F. Mezzadri , How to Generate Random Matrices from the Classical Compact Group . Not. Am. Math. Soc. 54 , 592 ( ). OpenUrl ↵ See supplementary materials. ↵ D. Porras , J. I. Cirac , Effective quantum spin systems with trapped ions . Phys. Rev. Lett. 92 , 207901 ( ). doi: 10.1103/PhysRevLett.92.207901 pmid: 15169383 OpenUrl ↵ P. Jurcevic , B. P. Lanyon , P. Hauke , C. Hempel , P. Zoller , R. Blatt , C. F. Roos , Quasiparticle engineering and entanglement propagation in a quantum many-body system . Nature 511 , 202 – 205 ( ). doi: 10.1038/nature13461 pmid: 25008526 OpenUrl CrossRef PubMed ↵ J. Smith , A. Lee , P. Richerme , B. Neyenhuis , P. W. Hess , P. Hauke , M. Heyl , D. A. Huse , C. Monroe , Many-body localization in a quantum simulator with programmable random disorder . Nat. Phys. 12 , 907 – 911 ( ). doi: 10.1038/nphys3783 OpenUrl CrossRef ↵ C. Maier , T. Brydges , P. Jurcevic , N. Trautmann , C. Hempel , P. Hauke , R. Blatt , C. F. Roos , Environment-assisted quantum transport in a 10-qubit network . Phys. Rev. Lett. 122 , 050501 ( ). doi: 10.1038/nphys3783 OpenUrl CrossRef ↵ M. P. Zaletel , R. S. K. Mong , C. Karrasch , J. E. Moore , F. Pollmann , Time-evolving a matrix product state with long-ranged interactions . Phys. Rev. B 91 , 165112 ( ). doi: 10.1103/PhysRevB.91.165112 OpenUrl CrossRef ↵ A. J. Daley , Quantum trajectories and open many-body quantum systems . Adv. Phys. 63 , 77 – 149 ( ). doi: 10.1080/00018732.2014.933502 OpenUrl CrossRef ↵ A. L. Burin , Localization in a random XY model with long-range interactions: Intermediate case between single-particle and many-body problems . Phys. Rev. B 92 , 104428 ( ). doi: 10.1103/PhysRevB.92.104428 OpenUrl ↵ M. Schreiber , S. S. Hodgman , P. Bordia , H. P. Lüschen , M. H. Fischer , R. Vosk , E. Altman , U. Schneider , I. Bloch , Observation of many-body localization of interacting fermions in a quasirandom optical lattice . Science 349 , 842 – 845 ( ). doi: 10.1126/science.aaa7432 pmid: 26229112 OpenUrl CrossRef PubMed ↵ J. H. Bardarson , F. Pollmann , J. E. Moore , Unbounded growth of entanglement in models of many-body localization . Phys. Rev. Lett. 109 , 017202 ( ). doi: 10.1103/PhysRevLett.109.017202 pmid: 23031128 OpenUrl CrossRef PubMed ↵ M. Serbyn , Z. Papić , D. A. Abanin , Universal slow growth of entanglement in interacting strongly disordered systems . Phys. Rev. Lett. 110 , 260601 ( ). doi: 10.1103/PhysRevLett.110.260601 pmid: 23848859 OpenUrl CrossRef PubMed ↵ A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, M. Greiner, Probing entanglement in a many-body-localized system. arxiv:1805.09819 [cond-mat.quant-gas] (13 Jun 2018). ↵ A. Safavi-Naini , M. L. Wall , O. L. Acevedo , A. M. Rey , R. M. Nandkishore , Quantum dynamics of disordered spin chains with power-law interactions. . Phys. Rev. A 99 , 033610 ( ). doi: 10.1214/aos/1176345462 OpenUrl CrossRef ↵ B. Efron , C. Stein , The Jackknife Estimate of Variance . Ann. Stat. 9 , 586 – 596 ( ). doi: 10.1214/aos/1176345462 OpenUrl CrossRef ↵ T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, Probing Rényi Entanglement Entropies via Randomized Measurements, Version 2, Zenodo (2018); https://doi.org/10.5281/zenodo.2527010. doi: 10.5281/zenodo.2527010 ↵ T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, TiffBrydges/Renyi_Entanglement_Entropy v0, Zenodo (2019); https://doi.org/10.5281/zenodo.2533875. doi: 10.5281/zenodo.2533875 OpenUrl CrossRef PubMed ↵ D. Gross , Y.-K. Liu , S. T. Flammia , S. Becker , J. Eisert , Quantum state tomography via compressed sensing . Phys. Rev. Lett. 105 , 150401 ( ). doi: 10.1103/PhysRevLett.105.150401 pmid: 21230876 OpenUrl CrossRef PubMed M. M. Wolf , F. Verstraete , M. B. Hastings , J. I. Cirac , Area laws in quantum systems: Mutual information and correlations . Phys. Rev. Lett. 100 , 070502 ( ). doi: 10.1103/PhysRevLett.100.070502 pmid: 18352531 OpenUrl CrossRef PubMed Web of Science J. R. Johansson , P. D. Nation , F. Nori , QuTiP 2: A Python framework for the dynamics of open quantum systems . Comput. Phys. Commun. 184 , 1234 – 1240 ( ). doi: 10.1016/j.cpc.2012.11.019 OpenUrl CrossRef ↵ B. Pirvu , V. Murg , J. I. Cirac , F. Verstraete , Matrix product operator representations . New J. Phys. 12 , 025012 ( ). doi: 10.1088/1367-2630/12/2/025012 OpenUrl CrossRef

Acknowledgments: Funding: We acknowledge funding from the ERC Synergy Grant UQUAM, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement no. 741541, the SFB FoQuS (FWF project no. F4016-N23), and QTFLAG–QuantERA. Also, the project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQUANS). Author contributions: P.Z. suggested the research topic, which was further developed by A.E., B.V., B.P.L., and C.F.R. A.E., B.V., and P.Z. developed the theoretical protocols. P.J., C.M., T.B., B.P.L., C.F.R., and R.B. contributed to the experimental setup. T.B., P.J., C.M., and C.F.R. performed the experiments. A.E., B.V., and C.F.R. analyzed the data and carried out numerical simulations. T.B., A.E., B.V., B.P.L., P.Z., and C.F.R. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript. Competing interests: There are no competing interests. Data and materials availability: All data are publicly availableon Zenodo ( We acknowledge funding from the ERC Synergy Grant UQUAM, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement no. 741541, the SFB FoQuS (FWF project no. F4016-N23), and QTFLAG–QuantERA. Also, the project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQUANS).P.Z. suggested the research topic, which was further developed by A.E., B.V., B.P.L., and C.F.R. A.E., B.V., and P.Z. developed the theoretical protocols. P.J., C.M., T.B., B.P.L., C.F.R., and R.B. contributed to the experimental setup. T.B., P.J., C.M., and C.F.R. performed the experiments. A.E., B.V., and C.F.R. analyzed the data and carried out numerical simulations. T.B., A.E., B.V., B.P.L., P.Z., and C.F.R. wrote the manuscript. All authors contributed to the discussion of the results and the manuscript.There are no competing interests.All data are publicly availableon Zenodo ( 41 ). All code used for data evaluation and numerical simulations is publicly available on Zenodo ( 42 ).