Quantum thermalization through entanglement in an isolated many-body system

Selective Detection of Dynamics-Complete Set of Correlations via Quantum Channels

Correlations of fluctuations are essential to understanding many-body systems and key information for advancing quantum technologies. To fully describe the dynamics of a physical system, all time-ordered correlations (TOCs), i.e., the dynamics-complete set of correlations are needed. The current measurement techniques can only access a limited set of TOCs, and there has been no systematic and feasible solution for extracting the dynamic-complete set of correlations hitherto. Here we propose a platform-universal protocol to selectively detect arbitrary types of TOCs via quantum channels. In our method, the quantum channels are synthesized with various controls, and engineer the evolution of a sensor-target system along a specific path that corresponds to a desired correlation. Using nuclear magnetic resonance, we experimentally demonstrate this protocol by detecting a specific type of fourth-order TOC that has never been accessed previously. We also show that the knowledge of the TOCs can be used to significantly improve the precision of quantum optimal control. Our method provides a new toolbox for characterizing the quantum many-body states and quantum noise, and hence for advancing the fields of quantum sensing and quantum computing.

An atomic boson sampler

A boson sampler implements a restricted model of quantum computing. It is defined by the ability to sample from the distribution resulting from the interference of identical bosons propagating according to programmable, non-interacting dynamics1. An efficient exact classical simulation of boson sampling is not believed to exist, which has motivated ground-breaking boson sampling experiments in photonics with increasingly many photons2-12. However, it is difficult to generate and reliably evolve specific numbers of photons with low loss, and thus probabilistic techniques for postselection7 or marked changes to standard boson sampling10-12 are generally used. Here, we address the above challenges by implementing boson sampling using ultracold atoms13,14 in a two-dimensional, tunnel-coupled optical lattice. This demonstration is enabled by a previously unrealized combination of tools involving high-fidelity optical cooling and imaging of atoms in a lattice, as well as programmable control of those atoms using optical tweezers. When extended to interacting systems, our work demonstrates the core abilities required to directly assemble ground and excited states in simulations of various Hubbard models15,16.

Probing entanglement in a 2D hard-core Bose-Hubbard lattice

Entanglement and its propagation are central to understanding many physical properties of quantum systems1-3. Notably, within closed quantum many-body systems, entanglement is believed to yield emergent thermodynamic behaviour4-7. However, a universal understanding remains challenging owing to the non-integrability and computational intractability of most large-scale quantum systems. Quantum hardware platforms provide a means to study the formation and scaling of entanglement in interacting many-body systems8-14. Here we use a controllable 4 × 4 array of superconducting qubits to emulate a 2D hard-core Bose-Hubbard (HCBH) lattice. We generate superposition states by simultaneously driving all lattice sites and extract correlation lengths and entanglement entropy across its many-body energy spectrum. We observe volume-law entanglement scaling for states at the centre of the spectrum and a crossover to the onset of area-law scaling near its edges.

Benchmarking highly entangled states on a 60-atom analogue quantum simulator

Quantum systems have entered a competitive regime in which classical computers must make approximations to represent highly entangled quantum states1,2. However, in this beyond-classically-exact regime, fidelity comparisons between quantum and classical systems have so far been limited to digital quantum devices2-5, and it remains unsolved how to estimate the actual entanglement content of experiments6. Here, we perform fidelity benchmarking and mixed-state entanglement estimation with a 60-atom analogue Rydberg quantum simulator, reaching a high-entanglement entropy regime in which exact classical simulation becomes impractical. Our benchmarking protocol involves extrapolation from comparisons against an approximate classical algorithm, introduced here, with varying entanglement limits. We then develop and demonstrate an estimator of the experimental mixed-state entanglement6, finding our experiment is competitive with state-of-the-art digital quantum devices performing random circuit evolution2-5. Finally, we compare the experimental fidelity against that achieved by various approximate classical algorithms, and find that only the algorithm we introduce is able to keep pace with the experiment on the classical hardware we use. Our results enable a new model for evaluating the ability of both analogue and digital quantum devices to generate entanglement in the beyond-classically-exact regime, and highlight the evolving divide between quantum and classical systems.

Estimating entanglement entropy via variational quantum circuits with classical neural networks

Entropy plays a crucial role in both physics and information science, encompassing classical and quantum domains. In this paper, we present the quantum neural entropy estimator (QNEE), an approach that combines classical neural network (NN) with variational quantum circuits to estimate the von Neumann and Rényi entropies of a quantum state. QNEE provides accurate estimates of entropy while also yielding the eigenvalues and eigenstates of the input density matrix. Leveraging the capabilities of classical NN, QNEE can classify different phases of quantum systems that accompany the changes of entanglement entropy. Our numerical simulation demonstrates the effectiveness of QNEE by applying it to the 1D XXZ Heisenberg model. In particular, QNEE exhibits high sensitivity in estimating entanglement entropy near the phase transition point. We expect that QNEE will serve as a valuable tool for quantum entropy estimation and phase classification.

Stable quantum-correlated many-body states through engineered dissipation

Engineered dissipative reservoirs have the potential to steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using up to 49 superconducting qubits, we prepared low-energy states of the transverse-field Ising model through coupling to dissipative auxiliary qubits. In one dimension, we observed long-range quantum correlations and a ground-state fidelity of 0.86 for 18 qubits at the critical point. In two dimensions, we found mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we explored transport in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors.

Many-body quantum chaos in mixtures of multiple species

We study spectral correlations in many-body quantum mixtures of fermions, bosons, and qubits with periodically kicked spreading and mixing of species. We take two types of mixing, namely, Jaynes-Cummings and Rabi, respectively, satisfying and breaking the conservation of a total number of species. We analytically derive the generating Hamiltonians whose spectral properties determine the spectral form factor in the leading order. We further analyze the system-size (L) scaling of Thouless time t^{*}, beyond which the spectral form factor follows the prediction of random matrix theory. The L dependence of t^{*} crosses over from lnL to L^{2} with an increasing Jaynes-Cummings mixing between qubits and fermions or bosons in a finite-size chain, and it finally settles to t^{*}∝O(L^{2}) in the thermodynamic limit for any mixing strength. The Rabi mixing between qubits and fermions leads to t^{*}∝O(lnL), previously predicted for single species of qubits or fermions without total-number conservation.

Mesoscopic fluctuations in entanglement dynamics

Understanding fluctuation phenomena plays a dominant role in the development of many-body physics. The time evolution of entanglement is essential to a broad range of subjects in many-body physics, ranging from exotic quantum matter to quantum thermalization. Stemming from various dynamical processes of information, fluctuations in entanglement evolution differ conceptually from out-of-equilibrium fluctuations of traditional physical quantities. Their studies remain elusive. Here we uncover an emergent random structure in the evolution of the many-body wavefunction in two classes of integrable-either interacting or noninteracting-lattice models. It gives rise to out-of-equilibrium entanglement fluctuations which fall into the paradigm of mesoscopic fluctuations of wave interference origin. Specifically, the entanglement entropy variance obeys a universal scaling law in each class, and the full distribution displays a sub-Gaussian upper and a sub-Gamma lower tail. These statistics are independent of both the system's microscopic details and the choice of entanglement probes, and broaden the class of mesoscopic universalities. They have practical implications for controlling entanglement in mesoscopic devices.

Logical quantum processor based on reconfigurable atom arrays

Suppressing errors is the central challenge for useful quantum computing1, requiring quantum error correction (QEC)2-6 for large-scale processing. However, the overhead in the realization of error-corrected 'logical' qubits, in which information is encoded across many physical qubits for redundancy2-4, poses substantial challenges to large-scale logical quantum computing. Here we report the realization of a programmable quantum processor based on encoded logical qubits operating with up to 280 physical qubits. Using logical-level control and a zoned architecture in reconfigurable neutral-atom arrays7, our system combines high two-qubit gate fidelities8, arbitrary connectivity7,9, as well as fully programmable single-qubit rotations and mid-circuit readout10-15. Operating this logical processor with various types of encoding, we demonstrate improvement of a two-qubit logic gate by scaling surface-code6 distance from d = 3 to d = 7, preparation of colour-code qubits with break-even fidelities5, fault-tolerant creation of logical Greenberger-Horne-Zeilinger (GHZ) states and feedforward entanglement teleportation, as well as operation of 40 colour-code qubits. Finally, using 3D [[8,3,2]] code blocks16,17, we realize computationally complex sampling circuits18 with up to 48 logical qubits entangled with hypercube connectivity19 with 228 logical two-qubit gates and 48 logical CCZ gates20. We find that this logical encoding substantially improves algorithmic performance with error detection, outperforming physical-qubit fidelities at both cross-entropy benchmarking and quantum simulations of fast scrambling21,22. These results herald the advent of early error-corrected quantum computation and chart a path towards large-scale logical processors.

Quantifying spatiotemporal patterns in classical and quantum systems out of equilibrium

A rich variety of nonequilibrium dynamical phenomena and processes unambiguously calls for the development of general numerical techniques to probe and estimate a complex interplay between spatial and temporal degrees of freedom in many-body systems of completely different nature. In this work we provide a solution to this problem by adopting a structural complexity measure to quantify spatiotemporal patterns in the time-dependent digital representation of a system. On the basis of very limited amount of data our approach allows us to distinguish different dynamical regimes and define critical parameters in both classical and quantum systems. By the example of the discrete time crystal realized in nonequilibrium quantum systems we provide a complete low-level characterization of this nontrivial dynamical phase with only processing bitstrings, which can be considered as a valuable alternative to previous studies based on the calculations of qubit correlation functions.

Classical route to ergodicity and scarring in collective quantum systems

Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems.

Finite-Time Teleportation Phase Transition in Random Quantum Circuits

How long does it take to entangle two distant qubits in a quantum circuit evolved by generic unitary dynamics? We show that if the time evolution is followed by measurements of all but two infinitely separated test qubits, then the entanglement between them can undergo a phase transition and become nonzero at a finite critical time t_{c}. The fidelity of teleporting a quantum state from an input qubit to an infinitely distant output qubit shows the same critical onset. Specifically, these finite-time transitions occur in short-range interacting two-dimensional random unitary circuits and in sufficiently long-range interacting one-dimensional circuits. The phase transition is understood by mapping the random continuous-time evolution to a finite-temperature thermal state of an effective spin Hamiltonian, where the inverse temperature equals the evolution time in the circuit. In this framework, the entanglement between two distant qubits at times t>t_{c} corresponds to the emergence of long-range ferromagnetic spin correlations below the critical temperature. We verify these predictions using numerical simulation of Clifford circuits and propose potential realizations in existing platforms for quantum simulation.

Commensurate and incommensurate 1D interacting quantum systems

Single-atom imaging resolution of many-body quantum systems in optical lattices is routinely achieved with quantum-gas microscopes. Key to their great versatility as quantum simulators is the ability to use engineered light potentials at the microscopic level. Here, we employ dynamically varying microscopic light potentials in a quantum-gas microscope to study commensurate and incommensurate 1D systems of interacting bosonic Rb atoms. Such incommensurate systems are analogous to doped insulating states that exhibit atom transport and compressibility. Initially, a commensurate system with unit filling and fixed atom number is prepared between two potential barriers. We deterministically create an incommensurate system by dynamically changing the position of the barriers such that the number of available lattice sites is reduced while retaining the atom number. Our systems are characterised by measuring the distribution of particles and holes as a function of the lattice filling, and interaction strength, and we probe the particle mobility by applying a bias potential. Our work provides the foundation for preparation of low-entropy states with controlled filling in optical-lattice experiments.

Particle-hole thermalization in a composite superconducting and normal-conducting nanowire

The mechanisms by which isolated condensed matter systems thermalize is a topic of growing interest. Thermalization is known to be linked to the emergence of chaos in the dynamics of a system. We show that a solid state scattering system, containing superconducting elements, can thermalize scattered states without affecting the degree of entanglement of the scattered states. We consider a composite NSNSNSNSN nanowire composed of Bi_{2}Sr_{2}CaCu_{2}O_{8+x} superconducting segments (S) and normal conducting segments (N). We consider parameter regimes where all current flow is due to tunneling currents that are facilitated by quasibound state resonances inside the SNSNSNS structure. At certain energies, scattered pure states approach ergodicity, even though they remain pure.

Unifying Emergent Hydrodynamics and Lindbladian Low-Energy Spectra across Symmetries, Constraints, and Long-Range Interactions

We identify emergent hydrodynamics governing charge transport in Brownian random time evolution with various symmetries, constraints, and ranges of interactions. This is accomplished via a mapping between the averaged dynamics and the low-energy spectrum of a Lindblad operator, which acts as an effective Hamiltonian in a doubled Hilbert space. By explicitly constructing dispersive excited states of this effective Hamiltonian using a single-mode approximation, we provide a comprehensive understanding of diffusive, subdiffusive, and superdiffusive relaxation in many-body systems with conserved multipole moments and variable interaction ranges. Our approach further allows us to identify exotic Krylov-space-resolved diffusive relaxation despite the presence of dipole conservation, which we verify numerically. Therefore, we provide a general and versatile framework to qualitatively understand the dynamics of conserved operators under random unitary time evolution.

Sunburst quantum Ising model under interaction quench: Entanglement and role of initial state coherence

We study the nonequilibrium dynamics of an isolated bipartite quantum system, the sunburst quantum Ising model, under interaction quench. The prequench limit of this model is two noninteracting integrable systems, namely a transverse Ising chain and finite number of isolated qubits. As a function of interaction strength, the spectral fluctuation property goes from Poisson to Wigner-Dyson statistics. We chose entanglement entropy as a probe to study the approach to thermalization or lack of it in postquench dynamics. In the near-integrable limit, as expected, the linear entropy displays oscillatory behavior, while in the chaotic limit it saturates. Along with the chaotic nature of the time evolution generator, we show the importance of the role played by the coherence of the initial state in deciding the nature of thermalization. We further show that these findings are general by replacing the Ising ring with a disordered XXZ model with disorder strength putting it in the many-body localized phase.

Entangling Dynamics from Effective Rotor-Spin-Wave Separation in U(1)-Symmetric Quantum Spin Models

The nonequilibrium dynamics of quantum spin models is a most challenging topic, due to the exponentiality of Hilbert space, and it is central to the understanding of the many-body entangled states that can be generated by state-of-the-art quantum simulators. A particularly important class of evolutions is the one governed by U(1)-symmetric Hamiltonians, initialized in a state that breaks the U(1) symmetry-the paradigmatic example being the evolution of the so-called one-axis-twisting (OAT) model, featuring infinite-range interactions between spins. In this Letter, we show that the dynamics of the OAT model can be closely reproduced by systems with power-law-decaying interactions, thanks to an effective separation between the zero-momentum degrees of freedom, associated with the so-called Anderson tower of states, and reconstructing an OAT model, as well as finite-momentum ones, associated with spin-wave excitations. This mechanism explains quantitatively the recent numerical observation of spin squeezing and Schrödinger cat-state generation in the dynamics of dipolar Hamiltonians, and it paves the way for the extension of this observation to a much larger class of models of immediate relevance for quantum simulations.

General Class of Continuous Variable Entanglement Criteria

We present a general class of entanglement criteria for continuous variable systems. Our criteria are based on the Husimi Q distribution and allow for optimization over the set of all concave functions rendering them extremely general and versatile. We show that several entropic criteria and second moment criteria are obtained as special cases. Our criteria reveal entanglement of families of states undetected by any commonly used criteria and provide clear advantages under typical experimental constraints such as finite detector resolution and measurement statistics.

One-dimensional Lévy quasicrystal

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights. We introduce Lévy quasicrystal by discretizing the space-fractional Schrödinger equation using the Grünwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrödinger equation maps to the Aubry-André (AA) Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between Lévy quasicrystal and the AA model with power-law hopping, and show that the Lévy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.

Benchmarking Quantum Simulators Using Ergodic Quantum Dynamics

We propose and analyze a sample-efficient protocol to estimate the fidelity between an experimentally prepared state and an ideal target state, applicable to a wide class of analog quantum simulators without advanced spatiotemporal control. Our protocol relies on universal fluctuations emerging from generic Hamiltonian dynamics, which we discover in the present work. It does not require fine-tuned control over state preparation, quantum evolution, or readout capability, while achieving near optimal sample complexity: a percent-level precision is obtained with ∼10^{3} measurements, independent of system size. Furthermore, the accuracy of our fidelity estimation improves exponentially with increasing system size. We numerically demonstrate our protocol in a variety of quantum simulator platforms, including quantum gas microscopes, trapped ions, and Rydberg atom arrays. We discuss applications of our method for tasks such as multiparameter estimation of quantum states and processes.

Realizing spin squeezing with Rydberg interactions in an optical clock

Neutral-atom arrays trapped in optical potentials are a powerful platform for studying quantum physics, combining precise single-particle control and detection with a range of tunable entangling interactions1-3. For example, these capabilities have been leveraged for state-of-the-art frequency metrology4,5 as well as microscopic studies of entangled many-particle states6-11. Here we combine these applications to realize spin squeezing-a widely studied operation for producing metrologically useful entanglement-in an optical atomic clock based on a programmable array of interacting optical qubits. In this demonstration of Rydberg-mediated squeezing with a neutral-atom optical clock, we generate states that have almost four decibels of metrological gain. In addition, we perform a synchronous frequency comparison between independent squeezed states and observe a fractional-frequency stability of 1.087(1) × 10-15 at one-second averaging time, which is 1.94(1) decibels below the standard quantum limit and reaches a fractional precision at the 10-17 level during a half-hour measurement. We further leverage the programmable control afforded by optical tweezer arrays to apply local phase shifts to explore spin squeezing in measurements that operate beyond the relative coherence time with the optical local oscillator. The realization of this spin-squeezing protocol in a programmable atom-array clock will enable a wide range of quantum-information-inspired techniques for optimal phase estimation and Heisenberg-limited optical atomic clocks12-16.

Real-time broadening of bath-induced density profiles from closed-system correlation functions

The Lindblad master equation is one of the main approaches to open quantum systems. While it has been widely applied in the context of condensed matter systems to study properties of steady states in the limit of long times, the actual route to such steady states has attracted less attention yet. Here, we investigate the nonequilibrium dynamics of spin chains with a local coupling to a single Lindblad bath and analyze the transport properties of the induced magnetization. Combining typicality and equilibration arguments with stochastic unraveling, we unveil for the case of weak driving that the dynamics in the open system can be constructed on the basis of correlation functions in the closed system, which establishes a connection between the Lindblad approach and linear response theory at finite times. In this way, we provide a particular example where closed and open approaches to quantum transport agree strictly. We demonstrate this fact numerically for the spin-1/2 XXZ chain at the isotropic point and in the easy-axis regime, where superdiffusive and diffusive scaling is observed, respectively.

Deep learning of quantum entanglement from incomplete measurements

The quantification of the entanglement present in a physical system is of paramount importance for fundamental research and many cutting-edge applications. Now, achieving this goal requires either a priori knowledge on the system or very demanding experimental procedures such as full state tomography or collective measurements. Here, we demonstrate that, by using neural networks, we can quantify the degree of entanglement without the need to know the full description of the quantum state. Our method allows for direct quantification of the quantum correlations using an incomplete set of local measurements. Despite using undersampled measurements, we achieve a quantification error of up to an order of magnitude lower than the state-of-the-art quantum tomography. Furthermore, we achieve this result using networks trained using exclusively simulated data. Last, we derive a method based on a convolutional network input that can accept data from various measurement scenarios and perform, to some extent, independently of the measurement device.

Quantum simulation of thermodynamics in an integrated quantum photonic processor

One of the core questions of quantum physics is how to reconcile the unitary evolution of quantum states, which is information-preserving and time-reversible, with evolution following the second law of thermodynamics, which, in general, is neither. The resolution to this paradox is to recognize that global unitary evolution of a multi-partite quantum state causes the state of local subsystems to evolve towards maximum-entropy states. In this work, we experimentally demonstrate this effect in linear quantum optics by simultaneously showing the convergence of local quantum states to a generalized Gibbs ensemble constituting a maximum-entropy state under precisely controlled conditions, while introducing an efficient certification method to demonstrate that the state retains global purity. Our quantum states are manipulated by a programmable integrated quantum photonic processor, which simulates arbitrary non-interacting Hamiltonians, demonstrating the universality of this phenomenon. Our results show the potential of photonic devices for quantum simulations involving non-Gaussian states.

Witnessing light-driven entanglement using time-resolved resonant inelastic X-ray scattering

Characterizing and controlling entanglement in quantum materials is crucial for the development of next-generation quantum technologies. However, defining a quantifiable figure of merit for entanglement in macroscopic solids is theoretically and experimentally challenging. At equilibrium the presence of entanglement can be diagnosed by extracting entanglement witnesses from spectroscopic observables and a nonequilibrium extension of this method could lead to the discovery of novel dynamical phenomena. Here, we propose a systematic approach to quantify the time-dependent quantum Fisher information and entanglement depth of transient states of quantum materials with time-resolved resonant inelastic x-ray scattering. Using a quarter-filled extended Hubbard model as an example, we benchmark the efficiency of this approach and predict a light-enhanced many-body entanglement due to the proximity to a phase boundary. Our work sets the stage for experimentally witnessing and controlling entanglement in light-driven quantum materials via ultrafast spectroscopic measurements.

Controlling Local Thermalization Dynamics in a Floquet-Engineered Dipolar Ensemble

Understanding the microscopic mechanisms of thermalization in closed quantum systems is among the key challenges in modern quantum many-body physics. We demonstrate a method to probe local thermalization in a large-scale many-body system by exploiting its inherent disorder and use this to uncover the thermalization mechanisms in a three-dimensional, dipolar-interacting spin system with tunable interactions. Utilizing advanced Hamiltonian engineering techniques to explore a range of spin Hamiltonians, we observe a striking change in the characteristic shape and timescale of local correlation decay as we vary the engineered exchange anisotropy. We show that these observations originate from the system's intrinsic many-body dynamics and reveal the signatures of conservation laws within localized clusters of spins, which do not readily manifest using global probes. Our method provides an exquisite lens into the tunable nature of local thermalization dynamics and enables detailed studies of scrambling, thermalization, and hydrodynamics in strongly interacting quantum systems.

Work Extraction from Unknown Quantum Sources

Energy extraction is a central task in thermodynamics. In quantum physics, ergotropy measures the amount of work extractable under cyclic Hamiltonian control. As its full extraction requires perfect knowledge of the initial state, however, it does not characterize the work value of unknown or untrusted quantum sources. Fully characterizing such sources would require quantum tomography, which is prohibitively costly in experiments due to the exponential growth of required measurements and operational limitations. Here, we therefore derive a new notion of ergotropy applicable when nothing is known about the quantum states produced by the source, apart from what can be learned by performing only a single type of coarse-grained measurement. We find that in this case the extracted work is defined by the Boltzmann and observational entropy in cases where the measurement outcomes are, or are not, used in the work extraction, respectively. This notion of ergotropy represents a realistic measure of extractable work, which can be used as the relevant figure of merit to characterize a quantum battery.

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

Quench-Probe Setup as Analyzer of Fractionalized Entanglement Spreading

We propose a novel spatially inhomogeneous setup for revealing quench-induced fractionalized excitations in entanglement dynamics. In this quench-probe setting, the region undergoing a quantum quench is tunnel coupled to a static region, the probe. Subsequently, the time-dependent entanglement signatures of a tunable subset of excitations propagating to the probe are monitored by energy selectivity. We exemplify the power of this generic approach by identifying a unique dynamical signature associated with the presence of an isolated Majorana zero mode in the postquench Hamiltonian. In this case excitations emitted from the topological part of the system give rise to a fractionalized jump of log(2)/2 in the entanglement entropy of the probe. This dynamical effect is highly sensitive to the localized nature of the Majorana zero mode, but does not require the preparation of a topological initial state.

Observation of entanglement transition of pseudo-random mixed states

Random quantum states serve as a powerful tool in various scientific fields, including quantum supremacy and black hole physics. It has been theoretically predicted that entanglement transitions may happen for different partitions of multipartite random quantum states; however, the experimental observation of these transitions is still absent. Here, we experimentally demonstrate the entanglement transitions witnessed by negativity on a fully connected superconducting processor. We apply parallel entangling operations, that significantly decrease the depth of the pseudo-random circuits, to generate pseudo-random pure states of up to 15 qubits. By quantum state tomography of the reduced density matrix of six qubits, we measure the negativity spectra. Then, by changing the sizes of the environment and subsystems, we observe the entanglement transitions that are directly identified by logarithmic entanglement negativities based on the negativity spectra. In addition, we characterize the randomness of our circuits by measuring the distance between the distribution of output bit-string probabilities and the Porter-Thomas distribution. Our results show that superconducting processors with all-to-all connectivity constitute a promising platform for generating random states and understanding the entanglement structure of multipartite quantum systems.

Entropy and temperature in finite isolated quantum systems

We investigate how the temperature calculated from the microcanonical entropy compares with the canonical temperature for finite isolated quantum systems. We concentrate on systems with sizes that make them accessible to numerical exact diagonalization. We thus characterize the deviations from ensemble equivalence at finite sizes. We describe multiple ways to compute the microcanonical entropy and present numerical results for the entropy and temperature computed in these various ways. We show that using an energy window whose width has a particular energy dependence results in a temperature with minimal deviations from the canonical temperature.

Many-Body Interference at the Onset of Chaos

We unveil the signature of many-body interference across dynamical regimes of the Bose-Hubbard model. Increasing the particles' indistinguishability enhances the temporal fluctuations of few-body observables, with a dramatic amplification at the onset of quantum chaos. By resolving the exchange symmetries of partially distinguishable particles, we explain this amplification as the fingerprint of the initial state's coherences in the eigenbasis.

Assigning temperatures to eigenstates

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.

Quantum coherence controls the nature of equilibration and thermalization in coupled chaotic systems

A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interaction with a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax toward an equilibrium. It is found that quantum coherence of the initial product states in the energy eigenbasis of the subsystems-quantified by the off-diagonal elements of the subsystem density matrices-can be viewed as a resource for equilibration and thermalization as manifested by the entanglement generated. Results are given for four distinct perturbation strength regimes, the ultraweak, weak, intermediate, and strong regimes. For each, three types of tensor product states are considered for the initial state: uniform superpositions, random superpositions, and individual subsystem eigenstates. A universal timescale is identified involving the interaction strength parameter. In particular, maximally coherent initial product states (a form of uniform superpositions) thermalize under time evolution for any perturbation strength in spite of the fact that in the ultraweak perturbative regime the underlying eigenstates of the system have a tensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity as the interaction vanishes. Moreover, it is shown that in the ultraweak regime the initial entanglement growth of the system whose initial states are maximally coherent is quadratic-in-time, in contrast to the widely observed linear behavior.

Exploring Integrability-Chaos Transition with a Sequence of Independent Perturbations

A gas of interacting particles is a paradigmatic example of chaotic systems. It is shown here that, even if all but one particle are fixed in generic positions, the excited states of the moving particle are chaotic. They are characterized by the number of principal components (NPC)-the number of integrable system eigenstates involved into the nonintegrable one, which increases linearly with the number of strong scatterers. This rule is a particular case of the general effect of an additional perturbation on the system chaotic properties. The perturbation independence criteria supposing the system chaoticity increase are derived here as well. The effect can be observed in experiments with photons or cold atoms as the decay of observable fluctuation variance, which is inversely proportional to NPC and, therefore, to the number of scatterers. This decay indicates that the eigenstate thermalization is approached. The results are confirmed by numerical calculations for a harmonic waveguide with zero-range scatterers along its axis.

Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU(2) symmetry

We investigate the eigenstate thermalization properties of the spin-1/2 XXZ model in two-dimensional rectangular lattices of size L_{1}×L_{2} under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size 4×7 and of the energy eigenstates up to 4×6. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional XXZ model follows the eigenstate thermalization hypothesis. When the spin interaction is isotropic, the XXZ model Hamiltonian conserves the total spin and has SU(2) symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace where the total spin is a good quantum number.

Preparing random states and benchmarking with many-body quantum chaos

Producing quantum states at random has become increasingly important in modern quantum science, with applications being both theoretical and practical. In particular, ensembles of such randomly distributed, but pure, quantum states underlie our understanding of complexity in quantum circuits¹ and black holes², and have been used for benchmarking quantum devices^3,4 in tests of quantum advantage^5,6. However, creating random ensembles has necessitated a high degree of spatio-temporal control^7-12 placing such studies out of reach for a wide class of quantum systems. Here we solve this problem by predicting and experimentally observing the emergence of random state ensembles naturally under time-independent Hamiltonian dynamics, which we use to implement an efficient, widely applicable benchmarking protocol. The observed random ensembles emerge from projective measurements and are intimately linked to universal correlations built up between subsystems of a larger quantum system, offering new insights into quantum thermalization¹³. Predicated on this discovery, we develop a fidelity estimation scheme, which we demonstrate for a Rydberg quantum simulator with up to 25 atoms using fewer than 10⁴ experimental samples. This method has broad applicability, as we demonstrate for Hamiltonian parameter estimation, target-state generation benchmarking, and comparison of analogue and digital quantum devices. Our work has implications for understanding randomness in quantum dynamics¹⁴ and enables applications of this concept in a much wider context^{4,5,9,10,15-20}.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Detecting Entanglement in Quantum Many-Body Systems via Permutation Moments

Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Because of the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this Letter, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have an effective implementation with either the generalized control-swap quantum circuits or the random unitary techniques. As an example, in the bipartite scenario, we develop an entanglement criterion that can detect bound entanglement and show strong detection capability in the multiqubit Ising model with a long-range XY Hamiltonian. In the multipartite case, the permutation-moment-based criteria can detect entangled states that are not detectable by any criteria extended from the bipartite case. Our framework also shows potential in entanglement quantification and entanglement structure detection.

Transitions in Computational Complexity of Continuous-Time Local Open Quantum Dynamics

We analyze the complexity of classically simulating continuous-time dynamics of locally interacting quantum spin systems with a constant rate of entanglement breaking noise. We prove that a polynomial time classical algorithm can be used to sample from the state of the spins when the rate of noise is higher than a threshold determined by the strength of the local interactions. Furthermore, by encoding a 1D fault tolerant quantum computation into the dynamics of spin systems arranged on two or higher dimensional grids, we show that for several noise channels, the problem of weakly simulating the output state of both purely Hamiltonian and purely dissipative dynamics is expected to be hard in the low-noise regime.

Chaos and bipartite entanglement between Bose-Josephson junctions

The entanglement between two weakly coupled bosonic Josephson junctions is studied in relation to the classical mixed phasespace structure of the system, containing symmetry-related regular islands separated by chaos. The symmetry-resolved entanglement spectrum and bipartite entanglement entropy of the system's energy eigenstates are calculated and compared to their expected structure for random states that exhibit complete or partial ergodicity. The entanglement spectra of chaos-supported eigenstates match the microcanonical structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic invariant that restricts ergodization on the energy shell. The symmetry-resolved entanglement entropy of these quasistochastic states consists of a mean-field maximum entanglement term and a fluctuation correction due to the finite size of the constituent subsystems. The total bipartite entanglement entropy of the eigenstates correlates with their chaoticity. Island-supported eigenstates are macroscopic Schrödinger cat states for particles and excitations with substantially lower entanglement.

Canonical Density Matrices from Eigenstates of Mixed Systems

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.

Erasure Tolerant Quantum Memory and the Quantum Null Energy Condition in Holographic Systems

Investigating principles for storage of quantum information at finite temperature with minimal need for active error correction is an active area of research. We bear upon this question in two-dimensional holographic conformal field theories via the quantum null energy condition that we have shown earlier to implement the restrictions imposed by quantum thermodynamics on such many-body systems. We study an explicit encoding of a logical qubit into two similar chirally propagating excitations of finite von Neumann entropy on a finite temperature background whose erasure can be implemented by an appropriate inhomogeneous and instantaneous energy-momentum inflow from an infinite energy memoryless bath due to which the system transits to a thermal state. Holographically, these fast erasure processes can be depicted by generalized AdS-Vaidya geometries described previously in which no assumption of specific form of bulk matter is needed. We show that the quantum null energy condition gives analytic results for the minimal finite temperature needed for the deletion which is larger than the initial background temperature in consistency with Landauer's principle. In particular, we find a simple expression for the minimum final temperature needed for the erasure of a large number of encoding qubits. We also find that if the encoding qubits are localized over an interval shorter than a specific localization length, then the fast erasure process is impossible, and furthermore this localization length is the largest for an optimal amount of encoding qubits determined by the central charge. We estimate the optimal encoding qubits for realistic protection against fast erasure. We discuss possible generalizations of our study for novel constructions of fault-tolerant quantum gates operating at finite temperature.

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

Ground-state energy distribution of disordered many-body quantum systems

Extreme-value distributions are studied in the context of a broad range of problems, from the equilibrium properties of low-temperature disordered systems to the occurrence of natural disasters. Our focus here is on the ground-state energy distribution of disordered many-body quantum systems. We derive an analytical expression that, upon tuning a parameter, reproduces with high accuracy the ground-state energy distribution of the systems that we consider. For some models, it agrees with the Tracy-Widom distribution obtained from Gaussian random matrices. They include transverse Ising models, the Sachdev-Ye model, and a randomized version of the PXP model. For other systems, such as Bose-Hubbard models with random couplings and the disordered spin-1/2 Heisenberg chain used to investigate many-body localization, the shapes are at odds with the Tracy-Widom distribution. Our analytical expression captures all of these distributions, thus playing a role to the lowest energy level similar to that played by the Brody distribution to the bulk of the spectrum.

Multipartite Entangled States in Dipolar Quantum Simulators

The scalable production of multipartite entangled states in ensembles of qubits is a crucial function of quantum devices, as such states are an essential resource both for fundamental studies on entanglement, as well as for applied tasks. Here we focus on the U(1) symmetric Hamiltonians for qubits with dipolar interactions-a model realized in several state-of-the-art quantum simulation platforms for lattice spin models, including Rydberg-atom arrays with resonant interactions. Making use of exact and variational simulations, we theoretically show that the nonequilibrium dynamics generated by this Hamiltonian shares fundamental features with that of the one-axis-twisting model, namely, the simplest interacting collective-spin model with U(1) symmetry. The evolution governed by the dipolar Hamiltonian generates a cascade of multipartite entangled states-spin-squeezed states, Schrödinger's cat states, and multicomponent superpositions of coherent spin states. Investigating systems with up to N=144 qubits, we observe full scalability of the entanglement features of these states directly related to metrology, namely, scalable spin squeezing at an evolution time O(N^{1/3}) and Heisenberg scaling of sensitivity of the spin parity to global rotations for cat states reached at times O(N). Our results suggest that the native Hamiltonian dynamics of state-of-the-art quantum simulation platforms, such as Rydberg-atom arrays, can act as a robust source of multipartite entanglement.

Statistical complexity and the road to equilibrium in many-body chaotic quantum systems

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.

Quantum gas microscope assisted with T-shape vacuum viewports

A quantum gas microscope plays an important role in cold-atom experiments, which provides a high-resolution imaging of the spatial distributions of cold atoms. Here we design, build and calibrate an integrated microscope for quantum gases with all the optical components fixed outside the vacuum chamber. It provides large numerical aperture (NA) of 0.75, as well as good optical access from side for atom loading in cold-atom experiments due to long working distance (7 mm fused silica+6 mm vacuum) of the microscope objective. We make a special design of the vacuum viewport with a T-shape window, to suppress the window flatness distortion introduced by the metal-glass binding process, and protect the high-resolution imaging from distortions due to unflattened window. The achieved Strehl ratio is 0.9204 using scanning-near-field microscopy (SNOM) fiber coupling incoherent light as point light source.

Scalable Spin Squeezing from Spontaneous Breaking of a Continuous Symmetry

Spontaneous symmetry breaking is a property of Hamiltonian equilibrium states which, in the thermodynamic limit, retain a finite average value of an order parameter even after a field coupled to it is adiabatically turned off. In the case of quantum spin models with continuous symmetry, we show that this adiabatic process is also accompanied by the suppression of the fluctuations of the symmetry generator-namely, the collective spin component along an axis of symmetry. In systems of S=1/2 spins or qubits, the combination of the suppression of fluctuations along one direction and of the persistence of transverse magnetization leads to spin squeezing-a much sought-after property of quantum states, both for the purpose of entanglement detection as well as for metrological uses. Focusing on the case of XXZ models spontaneously breaking a U(1) [or even SU(2)] symmetry, we show that the adiabatically prepared states have nearly minimal spin uncertainty; that the minimum phase uncertainty that one can achieve with these states scales as N^{-3/4} with the number of spins N; and that this scaling is attained after an adiabatic preparation time scaling linearly with N. Our findings open the door to the adiabatic preparation of strongly spin-squeezed states in a large variety of quantum many-body devices including, e.g., optical-lattice clocks.