Measuring entanglement entropy in a quantum many-body system

A quantum processor based on coherent transport of entangled atom arrays

The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is key for building scalable quantum information systems^1,2. In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation^3–5. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state^6,7. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits⁸ and a toric code state on a torus with sixteen data and eight ancillary qubits⁹. Finally, we use this architecture to realize a hybrid analogue–digital evolution² and use it for measuring entanglement entropy in quantum simulations^10–12, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars^13,14. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.

A quantum processer is realized using arrays of neutral atoms that are transported in a parallel manner by optical tweezers during computations, and used for quantum error correction and simulations.

Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement

In the context of ground states of quantum many-body systems, the locality of entanglement between connected regions of space is directly tied to the locality of the corresponding entanglement Hamiltonian: the latter is dominated by local, few-body terms. In this work, we introduce the negativity Hamiltonian as the (non-Hermitian) effective Hamiltonian operator describing the logarithm of the partial transpose of a many-body system. This allows us to address the connection between entanglement and operator locality beyond the paradigm of bipartite pure systems. As a first step in this direction, we study the structure of the negativity Hamiltonian for fermionic conformal field theories and a free-fermion chain: in both cases, we show that the negativity Hamiltonian assumes a quasilocal functional form, that is captured by simple functional relations.

Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

Using IBM's publicly accessible quantum computers, we have analyzed the entropies of Schrödinger's cat states, which have the form Ψ = (1/2)^1/2 [|0 0 0⋯0〉 + |1 1 1⋯1〉]. We have obtained the average Shannon entropy S_So of the distribution over measurement outcomes from 75 runs of 8192 shots, for each of the numbers of entangled qubits, on each of the quantum computers tested. For the distribution over N fault-free measurements on pure cat states, S_So would approach one as N → ∞, independent of the number of qubits; but we have found that S_So varies nearly linearly with the number of qubits n. The slope of S_Soversus the number of qubits differs among computers with the same quantum volumes. We have developed a two-parameter model that reproduces the near-linear dependence of the entropy on the number of qubits, based on the probabilities of observing the output 0 when a qubit is set to |0〉 and 1 when it is set to |1〉. The slope increases as the error rate increases. The slope provides a sensitive measure of the accuracy of a quantum computer, so it serves as a quickly determinable index of performance. We have used tomographic methods with error mitigation as described in the qiskit documentation to find the density matrix ρ and evaluate the von Neumann entropies of the cat states. From the reduced density matrices for individual qubits, we have calculated the entanglement entropies. The reduced density matrices represent mixed states with approximately 50/50 probabilities for states |0〉 and |1〉. The entanglement entropies are very close to one.

Tunable Geometries in Sparse Clifford Circuits

We investigate the emergence of different effective geometries in stochastic Clifford circuits with sparse coupling. By changing the probability distribution for choosing two-site gates as a function of distance, we generate sparse interactions that either decay or grow with distance as a function of a single tunable parameter. Tuning this parameter reveals three distinct regimes of geometry for the spreading of correlations and growth of entanglement in the system. We observe linear geometry for short-range interactions, treelike geometry on a sparse coupling graph for long-range interactions, and an intermediate fast scrambling regime at the crossover point between the linear and treelike geometries. This transition in geometry is revealed in calculations of the subsystem entanglement entropy and tripartite mutual information. We also study emergent lightcones that govern these effective geometries by teleporting a single qubit of information from an input qubit to an output qubit. These tools help to analyze distinct geometries arising in dynamics and correlation spreading in quantum many-body systems.

Entanglement Entropy in the Ising Model with Topological Defects

Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.

Cold atoms meet lattice gauge theory

The central idea of this review is to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by the fact that bosons are more 'accessible' and easier to manipulate for experimentalists, but this 'substitution' also leads to new physics and novel phenomena. It allows us to gain new information about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or [Formula: see text] Bose-Hubbard models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the Creutz-Hubbard ladder, or Gross-Neveu-Wilson and Wilson-Hubbard models. This article is not a general review of the rapidly growing field-it reviews activities related to quantum simulations for lattice field theories performed by the Quantum Optics Theory group at ICFO and their collaborators from 19 institutions all over the world. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant for particle physics. This article is part of the theme issue 'Quantum technologies in particle physics'.

Quantum register of fermion pairs

Quantum control of motion is central for modern atomic clocks¹ and interferometers². It enables protocols to process and distribute quantum information^3,4, and allows the probing of entanglement in correlated states of matter⁵. However, the motional coherence of individual particles can be fragile to maintain, as external degrees of freedom couple strongly to the environment. Systems in nature with robust motional coherence instead often involve pairs of particles, from the electrons in helium, to atom pairs⁶, molecules⁷ and Cooper pairs. Here we demonstrate long-lived motional coherence and entanglement of pairs of fermionic atoms in an optical lattice array. The common and relative motion of each pair realize a robust qubit, protected by exchange symmetry. The energy difference between the two motional states is set by the atomic recoil energy, is dependent on only the mass and the lattice wavelength, and is insensitive to the noise of the confining potential. We observe quantum coherence beyond ten seconds. Modulation of the interactions between the atoms provides universal control of the motional qubit. The methods presented here will enable coherently programmable quantum simulators of many-fermion systems⁸, precision metrology based on atom pairs and molecules^9,10 and, by implementing further advances^11-13, digital quantum computation using fermion pairs¹⁴.

Quantum and Classical Temporal Correlations in (1+1)D Quantum Cellular Automata

We employ (1+1)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the "present" and the "past." We show how the uniquely quantum part of these correlations-the coherence-can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.

Non-adiabatic dynamics of the entanglement entropy in a symmetry-breaking Haldane insulator

We study the non-adiabatic dynamics of a typical symmetry-protected topological (SPT) phase-the Haldane insulator (HI) phase with broken bond-centered inversion. By continuously breaking the middle chain, we find the gap closes at a critical point in the deep HI regime with a change of particle number partition of the left or right system. The adiabatic evolution fails at this critical point and we show how to predict the dynamics of the entanglement entropy near this point using a two-level model. These results show that one can find a critical regime where the entanglement measurement is relatively robust against perturbation that breaks the protecting symmetries in the HI. This is in contrast to the common belief that the SPT phases are fragile without the protecting symmetries.

Nonequilibrium Dynamics of Renyi Entropy for Bosonic Many-Particle Systems

We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.

Observation of fragmentation of a spinor Bose-Einstein condensate

[Figure: see text].

Quantum Simulation of the Bosonic Creutz Ladder with a Parametric Cavity

There has been a growing interest in realizing quantum simulators for physical systems where perturbative methods are ineffective. The scalability and flexibility of circuit quantum electrodynamics make it a promising platform for implementing various types of simulators, including lattice models of strongly coupled field theories. Here, we use a multimode superconducting parametric cavity as a hardware-efficient analog quantum simulator, realizing a lattice in synthetic dimensions with complex hopping interactions. The coupling graph, i.e., the realized model, can be programmed in situ. The complex-valued hopping interaction further allows us to simulate, for instance, gauge potentials and topological models. As a demonstration, we simulate a plaquette of the bosonic Creutz ladder. We characterize the lattice with scattering measurements, reconstructing the experimental Hamiltonian and observing important precursors of topological features including nonreciprocal transport and Aharonov-Bohm caging. This platform can be easily extended to larger lattices and different models involving other interactions.

Lieb-Robinson Bound and Almost-Linear Light Cone in Interacting Boson Systems

In this work, we investigate how quickly local perturbations propagate in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, these systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on a specific but experimentally natural situation in which the number of bosons at any one site in the unperturbed initial state is approximately limited. We rigorously prove the existence of an almost-linear information-propagation light cone, thus establishing a Lieb-Robinson bound: the wave front grows at most as t log^{2}(t). We prove the clustering theorem for gapped ground states and study the time complexity of classically simulating one-dimensional quench dynamics, a topic of great practical interest.

Quantifying and Controlling Entanglement in the Quantum Magnet Cs_{2}CoCl_{4}

The lack of methods to experimentally detect and quantify entanglement in quantum matter impedes our ability to identify materials hosting highly entangled phases, such as quantum spin liquids. We thus investigate the feasibility of using inelastic neutron scattering (INS) to implement a model-independent measurement protocol for entanglement based on three entanglement witnesses: one-tangle, two-tangle, and quantum Fisher information (QFI). We perform high-resolution INS measurements on Cs_{2}CoCl_{4}, a close realization of the S=1/2 transverse-field XXZ spin chain, where we can control entanglement using the magnetic field, and compare with density-matrix renormalization group calculations for validation. The three witnesses allow us to infer entanglement properties and make deductions about the quantum state in the material. We find QFI to be a particularly robust experimental probe of entanglement, whereas the one and two-tangles require more careful analysis. Our results lay the foundation for a general entanglement detection protocol for quantum spin systems.

Entanglement Quantification in Atomic Ensembles

Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500  atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.

Information-theoretic measures of superconductivity in a two-dimensional doped Mott insulator

A key open issue in condensed-matter physics is how quantum and classical correlations emerge in an unconventional superconductor from the underlying normal state. We study this problem in a doped Mott insulator with information-theory tools on the two-dimensional (2D) Hubbard model at finite temperature with cluster dynamical mean-field theory. We find that the local entropy detects the superconducting state and that the difference in the local entropy between the superconducting and normal states follows the same difference in the potential energy. We find that the thermodynamic entropy is suppressed in the superconducting state and monotonically decreases with decreasing doping. The maximum in entropy found in the normal state above the overdoped region of the superconducting dome is obliterated by superconductivity. The total mutual information, which quantifies quantum and classical correlations, is amplified in the superconducting state of the doped Mott insulator for all doping levels and shows a broad peak versus doping, as a result of competing quantum and classical effects.

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state's angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies-Thakkar, Lieb-Thirring, Redheffer-Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.

Experimentally Accessible Bounds on Distillable Entanglement from Entropic Uncertainty Relations

Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them to physical models realizable in cold-atom experiments. The derived entanglement bounds rely on measurements in only two different bases and are generically applicable to any quantum simulation platform.

High-powered optical superlattice with robust phase stability for quantum gas microscopy

Optical superlattice has a wide range of applications in the study of ultracold atom physics. Especially, it can be used to trap and manipulate thousands of atom pairs in parallel which constitutes a promising system for quantum simulation and quantum computation. In the present work, we report on a high-power optical superlattice formed by a 532-nm and 1064-nm dual-wavelength interferometer with a short lattice spacing of 630 nm. The short-term fluctuation (in 10 seconds) of the relative phase between the short lattice and the long lattice is measured to be 0.003π, which satisfies the needs for performing two-qubit gates among neighboring lattice sites. We further implement this superlattice in a ⁸⁷Rb experiment with a quantum gas microscope of single-site resolution, where the high-power 532-nm laser is necessary for pinning atoms in the short lattice during imaging, providing a unique platform for engineering quantum states.

Exact Thermalization Dynamics in the "Rule 54" Quantum Cellular Automaton

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as "Rule 54." For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulas for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical nonequilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.

Topological Lower Bound on Quantum Chaos by Entanglement Growth

A fundamental result in modern quantum chaos theory is the Maldacena-Shenker-Stanford upper bound on the growth of out-of-time-order correlators, whose infinite-temperature limit is related to the operator-space entanglement entropy of the evolution operator. Here we show that, for one-dimensional quantum cellular automata (QCA), there exists a lower bound on quantum chaos quantified by such entanglement entropy. This lower bound is equal to twice the index of the QCA, which is a topological invariant that measures the chirality of information flow, and holds for all the Rényi entropies, with its strongest Rényi-∞ version being tight. The rigorous bound rules out the possibility of any sublinear entanglement growth behavior, showing in particular that many-body localization is forbidden for unitary evolutions displaying nonzero index. Since the Rényi entropy is measurable, our findings have direct experimental relevance. Our result is robust against exponential tails which naturally appear in quantum dynamics generated by local Hamiltonians.

Chaos and Ergodicity across the Energy Spectrum of Interacting Bosons

We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.

Entropy in the Non-Fermi-Liquid Regime of the Doped 2D Hubbard Model

We study thermodynamic properties of the doped Hubbard model on the square lattice in the regime of strong charge and spin fluctuations at low temperatures near the metal-to-insulator crossover and obtain results with controlled accuracy using the diagrammatic Monte Carlo method directly in the thermodynamic limit. The behavior of the entropy reveals a non-Fermi-liquid state at sufficiently high interactions near half filling: A maximum in the entropy at nonzero doping develops as the coupling strength is increased, along with an inflection point, evidencing a metal to non-Fermi-liquid crossover. The specific heat exhibits additional distinctive features of a non-Fermi-liquid state. Measurements of the entropy can, therefore, be used as a probe of the state of the system in quantum simulation experiments with ultracold atoms in optical lattices.

Postselection-Free Entanglement Dynamics via Spacetime Duality

The dynamics of entanglement in "hybrid" nonunitary circuits (for example, involving both unitary gates and quantum measurements) has recently become an object of intense study. A major hurdle toward experimentally realizing this physics is the need to apply postselection on random measurement outcomes in order to repeatedly prepare a given output state, resulting in an exponential overhead. We propose a method to sidestep this issue in a wide class of nonunitary circuits by taking advantage of spacetime duality. This method maps the purification dynamics of a mixed state under nonunitary evolution onto a particular correlation function in an associated unitary circuit. This translates to an operational protocol which could be straightforwardly implemented on a digital quantum simulator. We discuss the signatures of different entanglement phases, and demonstrate examples via numerical simulations. With minor modifications, the proposed protocol allows measurement of the purity of arbitrary subsystems, which could shed light on the properties of the quantum error correcting code formed by the mixed phase in this class of hybrid dynamics.

Many-Body Chern Number from Statistical Correlations of Randomized Measurements

One of the main topological invariants that characterizes several topologically ordered phases is the many-body Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant are conventionally based on the linear-response techniques that require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancilla-free experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function. Remarkably, our results apply to disklike geometries that are more amenable to current quantum simulator architectures.

Correlation minor norms, entanglement detection and discord

In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord. We demonstrate that the CMN may be used to define a new measure for quantum discord.

Certifying the Adiabatic Preparation of Ultracold Lattice Bosons in the Vicinity of the Mott Transition

We present a joint experimental and theoretical analysis to assess the adiabatic experimental preparation of ultracold bosons in optical lattices aimed at simulating the three-dimensional Bose-Hubbard model. Thermometry of lattice gases is realized from the superfluid to the Mott regime by combining the measurement of three-dimensional momentum-space densities with ab initio quantum Monte Carlo (QMC) calculations of the same quantity. The measured temperatures are in agreement with isentropic lines reconstructed via QMC for the experimental parameters of interest, with a conserved entropy per particle of S/N=0.8(1)k_{B}. In addition, the Fisher information associated with this thermometry method shows that the latter is most accurate in the critical regime close to the Mott transition, as confirmed in the experiment. These results prove that equilibrium states of the Bose-Hubbard model-including those in the quantum-critical regime above the Mott transition-can be adiabatically prepared in cold-atom apparatus.

Quantum phases of Rydberg atoms on a kagome lattice

We analyze the zero-temperature phases of an array of neutral atoms on the kagome lattice, interacting via laser excitation to atomic Rydberg states. Density-matrix renormalization group calculations reveal the presence of a wide variety of complex solid phases with broken lattice symmetries. In addition, we identify a regime with dense Rydberg excitations that has a large entanglement entropy and no local order parameter associated with lattice symmetries. From a mapping to the triangular lattice quantum dimer model, and theories of quantum phase transitions out of the proximate solid phases, we argue that this regime could contain one or more phases with topological order. Our results provide the foundation for theoretical and experimental explorations of crystalline and liquid states using programmable quantum simulators based on Rydberg atom arrays.

Information Scrambling over Bipartitions: Equilibration, Entropy Production, and Typicality

In recent years, the out-of-time-order correlator (OTOC) has emerged as a diagnostic tool for information scrambling in quantum many-body systems. Here, we present exact analytical results for the OTOC for a typical pair of random local operators supported over two regions of a bipartition. Quite remarkably, we show that this "bipartite OTOC" is equal to the operator entanglement of the evolution, and we determine its interplay with entangling power. Furthermore, we compute long-time averages of the OTOC and reveal their connection with eigenstate entanglement. For Hamiltonian systems, we uncover a hierarchy of constraints over the structure of the spectrum and elucidate how this affects the equilibration value of the OTOC. Finally, we provide operational significance to this bipartite OTOC by unraveling intimate connections with average entropy production and scrambling of information at the level of quantum channels.

Towards future physics and applications via two-dimensional material NEMS resonators

Two-dimensional materials (2Dm) offer a unique insight into the world of quantum mechanics including van der Waals (vdWs) interactions, exciton dynamics and various other nanoscale phenomena. 2Dm are a growing family consisting of graphene, hexagonal-Boron Nitride (h-BN), transition metal dichalcogenides (TMDs), monochalcogenides (MNs), black phosphorus (BP), MXenes and 2D organic crystals such as small molecules (e.g., pentacene, C8 BTBT, perylene derivatives, etc.) and polymers (e.g., COF and MOF, etc.). They exhibit unique mechanical, electrical, optical and optoelectronic properties that are highly enhanced as the surface to volume ratio increases, resulting from the transition of bulk to the few- to mono- layer limit. Such unique attributes include the manifestation of highly tuneable bandgap semiconductors, reduced dielectric screening, highly enhanced many body interactions, the ability to withstand high strains, ferromagnetism, piezoelectric and flexoelectric effects. Using 2Dm for mechanical resonators has become a promising field in nanoelectromechanical systems (NEMS) for applications involving sensors and condensed matter physics investigations. 2Dm NEMS resonators react with their environment, exhibit highly nonlinear behaviour from tension induced stiffening effects and couple different physics domains. The small size and high stiffness of these devices possess the potential of highly enhanced force sensitivities for measuring a wide variety of un-investigated physical forces. This review highlights current research in 2Dm NEMS resonators from fundamental physics and an applications standpoint, as well as presenting future possibilities using these devices.

Unsupervised Machine Learning of Quantum Phase Transitions Using Diffusion Maps

Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method may work for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.

Quasiparticle Mass Enhancement as a Measure of Entanglement in the Kondo Problem

We analyze the quantum entanglement between opposite spin projection electrons in the ground state of the Anderson impurity model. In this model, a single level impurity with intralevel repulsion U is tunnel coupled to a free electron gas. The Anderson model presents a strongly correlated many body ground state with mass enhanced quasiparticle excitations. We find, using both analytical and numerical tools, that the quantum entanglement between opposite spin projection electrons is a monotonic universal function of the quasiparticle mass enhancement Z in the Kondo regime. This indicates that the interaction induced mass enhancement, which is generally used to quantify correlations in quantum many body systems, could be used as a measure of entanglement in the Kondo problem.

Mixed-State Entanglement from Local Randomized Measurements

We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].

Single-Copies Estimation of Entanglement Negativity

Entanglement plays a central role in quantum information processing and quantum physics. However, few effective ways are known to detect the amount of entanglement of an unknown quantum state. Here, we propose a scheme to estimate the entanglement negativity for any bipartition of a composite system. The proposed scheme is based on the random unitary evolution and local measurements on a single-copy quantum state, which is more practical compared to former methods based on collective measurements on many copies of the identical state. Meanwhile, we generalize the scheme to quantify the total correlation. We demonstrate the efficiency of the scheme with statistical analyses and numerical simulations. Our scheme is quite suitable for state-of-the-art quantum platforms, which can serve as a useful benchmarking tool to advance quantum technologies and a probe to study fundamental quantum physics like entanglement dynamics.

Measurement of Identical Particle Entanglement and the Influence of Antisymmetrization

We explore the relationship between symmetrization and entanglement through measurements on few-particle systems in a multiwell potential. In particular, considering two or three trapped atoms, we measure and distinguish correlations arising from two different physical origins: antisymmetrization of the fermionic wave function and interaction between particles. We quantify this through the entanglement negativity of states, and the introduction of an antisymmetric negativity, which allows us to understand the role that symmetrization plays in the measured entanglement properties. We apply this concept both to pure theoretical states and to experimentally reconstructed density matrices of two or three mobile particles in an array of optical tweezers.

Identification of Symmetry-Protected Topological States on Noisy Quantum Computers

Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely, symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum information processing to the classification of SPT phases of matter. They invoke the creation of a cluster state and are implemented on an IBM quantum computer. The experimental findings are compared to noisy simulations, allowing us to study the stability of topological states to perturbations and noise.

Area law of noncritical ground states in 1D long-range interacting systems

The area law for entanglement provides one of the most important connections between information theory and quantum many-body physics. It is not only related to the universality of quantum phases, but also to efficient numerical simulations in the ground state. Various numerical observations have led to a strong belief that the area law is true for every non-critical phase in short-range interacting systems. However, the area law for long-range interacting systems is still elusive, as the long-range interaction results in correlation patterns similar to those in critical phases. Here, we show that for generic non-critical one-dimensional ground states with locally bounded Hamiltonians, the area law robustly holds without any corrections, even under long-range interactions. Our result guarantees an efficient description of ground states by the matrix-product state in experimentally relevant long-range systems, which justifies the density-matrix renormalization algorithm.

The entanglement in non-critical ground states is conjectured to obey the area law, which is believed to arise from the short-range nature of interactions. Here the authors prove that the entanglement area law rigorously holds in one-dimensional systems even in the presence of long-range interactions.

Condition on the Rényi Entanglement Entropy under Stochastic Local Manipulation

The Rényi entanglement entropy (REE) is an entanglement quantifier considered as a natural generalization of the entanglement entropy. When it comes to stochastic local operations and classical communication (SLOCC), however, only a limited class of the REEs satisfy the monotonicity condition, while their statistical properties beyond mean values have not been fully investigated. Here, we establish a general condition that the probability distribution of the REE of any order obeys under SLOCC. The condition is obtained by introducing a family of entanglement monotones that contain the higher-order moments of the REEs. The contribution from the higher-order moments imposes a strict limitation on entanglement distillation via SLOCC. We find that the upper bound on success probabilities for entanglement distillation exponentially decreases as the amount of raised entanglement increases, which cannot be captured from the monotonicity of the REE. Based on the strong restriction on entanglement transformation under SLOCC, we design a new method to estimate entanglement in quantum many-body systems from experimentally observable quantities.

Energy redistribution and spatiotemporal evolution of correlations after a sudden quench of the Bose-Hubbard model

An optical lattice quantum simulator is an ideal experimental platform to investigate nonequilibrium dynamics of a quantum many-body system, which is, in general, hard to simulate with classical computers. Here, we use our quantum simulator of the Bose-Hubbard model to study dynamics far from equilibrium after a quantum quench. We successfully confirm the energy conservation law in the one- and three-dimensional systems and extract the propagation velocity of the single-particle correlation in the one- and two-dimensional systems. We corroborate the validity of our quantum simulator through quantitative comparisons between the experiments and the exact numerical calculations in one dimension. In the computationally hard cases of two or three dimensions, by using the quantum-simulation results as references, we examine the performance of a numerical method, namely, the truncated Wigner approximation, revealing its usefulness and limitation. This work constitutes an exemplary case for the usage of analog quantum simulators.

Quantum chaos in a system with high degree of symmetries

We study dynamical signatures of quantum chaos in one of the most relevant models in many-body quantum mechanics, the Bose-Hubbard model, whose high degree of symmetries yields a large number of invariant subspaces and degenerate energy levels. The standard procedure to reveal signatures of quantum chaos requires classifying the energy levels according to their symmetries, which may be experimentally and theoretically challenging. We show that this classification is not necessary to observe manifestations of spectral correlations in the temporal evolution of the survival probability, which makes this quantity a powerful tool in the identification of chaotic many-body quantum systems.

Robust Bilayer Charge Pumping for Spin- and Density-Resolved Quantum Gas Microscopy

Quantum gas microscopy has emerged as a powerful new way to probe quantum many-body systems at the microscopic level. However, layered or efficient spin-resolved readout methods have remained scarce as they impose strong demands on the specific atomic species and constrain the simulated lattice geometry and size. Here we present a novel high-fidelity bilayer readout, which can be used for full spin- and density-resolved quantum gas microscopy of two-dimensional systems with arbitrary geometry. Our technique makes use of an initial Stern-Gerlach splitting into adjacent layers of a highly stable vertical superlattice and subsequent charge pumping to separate the layers by 21  μm. This separation enables independent high-resolution images of each layer. We benchmark our method by spin- and density-resolving two-dimensional Fermi-Hubbard systems. Our technique furthermore enables the access to advanced entropy engineering schemes, spectroscopic methods, or the realization of tunable bilayer systems.

Fractionalized Excitations Revealed by Entanglement Entropy

Fractionalized excitations develop in many unusual many-body states such as quantum spin liquids, disordered phases that cannot be described using any local order parameter. Because these exotic excitations correspond to emergent degrees of freedom, how to probe them and establish their existence is a long-standing challenge. We present a general procedure to reveal the fractionalized excitations using real-space entanglement entropy in critical spin liquids that are particularly relevant to experiments. Moreover, we show how to use the entanglement entropy to construct the corresponding spinon Fermi surface. Our work defines a new pathway to establish and characterize exotic excitations in novel quantum phases of matter.

Multifractality Meets Entanglement: Relation for Nonergodic Extended States

In this work, we establish a relation between entanglement entropy and fractal dimension D of generic many-body wave functions, by generalizing the result of Page [Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291] to the case of sparse random pure states (SRPS). These SRPS living in a Hilbert space of size N are defined as normalized vectors with only N^{D} (0≤D≤1) random nonzero elements. For D=1, these states used by Page represent ergodic states at an infinite temperature. However, for 0<D<1, the SRPS are nonergodic and fractal, as they are confined in a vanishing ratio N^{D}/N of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy S_{1}(A) of a subsystem A, with Hilbert space dimension N_{A}, scales as S_{1}[over ¯](A)∼DlnN for small fractal dimensions D, N^{D}<N_{A}. Remarkably, S_{1}[over ¯](A) saturates at its thermal (Page) value at an infinite temperature, S_{1}[over ¯](A)∼lnN_{A} at larger D. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly nonergodic. Finally, we generalize our results to Renyi entropies S_{q}(A) with q>1 and to genuine multifractal states and also show that their fluctuations have ergodic behavior in a narrower vicinity of the ergodic state D=1.

Generation and manipulation of Schrödinger cat states in Rydberg atom arrays

Quantum entanglement involving coherent superpositions of macroscopically distinct states is among the most striking features of quantum theory, but its realization is challenging because such states are extremely fragile. Using a programmable quantum simulator based on neutral atom arrays with interactions mediated by Rydberg states, we demonstrate the creation of "Schrödinger cat" states of the Greenberger-Horne-Zeilinger (GHZ) type with up to 20 qubits. Our approach is based on engineering the energy spectrum and using optimal control of the many-body system. We further demonstrate entanglement manipulation by using GHZ states to distribute entanglement to distant sites in the array, establishing important ingredients for quantum information processing and quantum metrology.

Rydberg Atom Entanglements in the Weak Coupling Regime

We present an entanglement scheme for Rydberg atoms using the van der Waals interaction phase induced by Ramsey-type pulsed interactions. This scheme realizes not only controlled phase operations between atoms at a distance larger than Rydberg blockade distance, but also various counterintuitive entanglement examples, including two-atom entanglement in the presence of a closer third atom and W-state generation for three partially blockaded atoms. Experimental realization is conducted with single rubidium atoms in optical tweezer dipole traps, to demonstrate the proposed entanglement generations with an entanglement fidelity of F=0.59±0.11.

Measuring a dynamical topological order parameter in quantum walks

Quantum processes of inherent dynamical nature, such as quantum walks, defy a description in terms of an equilibrium statistical physics ensemble. Until now, identifying the general principles behind the underlying unitary quantum dynamics has remained a key challenge. Here, we show and experimentally observe that split-step quantum walks admit a characterization in terms of a dynamical topological order parameter (DTOP). This integer-quantized DTOP measures, at a given time, the winding of the geometric phase accumulated by the wavefunction during a quantum walk. We observe distinct dynamical regimes in our experimentally realized quantum walks, and each regime can be attributed to a qualitatively different temporal behavior of the DTOP. Upon identifying an equivalent many-body problem, we reveal an intriguing connection between the nonanalytic changes of the DTOP in quantum walks and the occurrence of dynamical quantum phase transitions.

Preparing Atomic Topological Quantum Matter by Adiabatic Nonunitary Dynamics

Motivated by the outstanding challenge of realizing low-temperature states of quantum matter in synthetic materials, we propose and study an experimentally feasible protocol for preparing topological states such as Chern insulators. By definition, such (nonsymmetry protected) topological phases cannot be attained without going through a phase transition in a closed system, largely preventing their preparation in coherent dynamics. To overcome this fundamental caveat, we propose to couple the target system to a conjugate system, so as to prepare a symmetry protected topological phase in an extended system by intermittently breaking the protecting symmetry. Finally, the decoupled conjugate system is discarded, thus projecting onto the desired topological state in the target system. By construction, this protocol may be immediately generalized to the class of invertible topological phases, characterized by the existence of an inverse topological order. We illustrate our findings with microscopic simulations on an experimentally realistic Chern insulator model of ultracold fermionic atoms in a driven spin-dependent hexagonal optical lattice.

Cross-Platform Verification of Intermediate Scale Quantum Devices

We describe a protocol for cross-platform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr[ρ_{1}ρ_{2}] and the purities Tr[ρ_{1,2}^{2}], and thus a fidelity of two, possibly mixed, quantum states ρ_{1} and ρ_{2} 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.

Spin-momentum entanglement in a Bose–Einstein condensate

Mechanisms including two types of Raman laser coupling (Ω₁ & Ω₂) and rf field coupling (Ω_rf) are applied to drive transitions between different hyperfine spin states. We investigated the entanglement between the spin and momentum degrees of freedom.