Expectation Values Of Local Observables Research Articles

Connected correlations in partitioning protocols: A case study and beyond

The assumption of local relaxation in inhomogeneous quantum quenches allows to compute asymptotically the expectation value of local observables via hydrodynamic arguments known as generalized hydrodynamics (GHD). In this work we address formally the question of when an observable is local enough to be described by GHD using the playground of partitioning protocols and non-interacting time evolution. We show that any state evolving under a quadratic Hamiltonian can be described via a set of decoupled dynamical fields such that one of those fields can be identified with a space-time-dependent generalisation of the root density. By studying the contribution to a connected spin correlation of each of those fields independently, we derive the locality conditions under which an observable can be described using the root density only. That shows both the regime of validity for hydrodynamic approaches that aim at describing the asymptotic value of observables in term of the root density only, such as GHD, and the locality conditions necessary for Gaussianification to occur.

Exact mean-field solution of a spin chain with short-range and long-range interactions

We consider the transverse field Ising model with additional all-to-all interactions between the spins. We show that a mean-field treatment of this model becomes exact in the thermodynamic limit, despite the presence of 1D short-range interactions. Namely, we show that the eigenstates of the model are coherent states with an amplitude that varies through the Hilbert space, within which expectation values of local observables can be computed with mean-field theory. We study then the thermodynamics of the model and identify the different phases. Among its peculiar features, this 1D model possesses a second-order phase transition at finite temperature and exhibits inverse melting.

On classical and hybrid shadows of quantum states

Classical shadows are a computationally efficient approach to storing quantum states on a classical computer for the purposes of estimating expectation values of local observables, obtained by performing repeated random measurements. In this note we offer some comments on this approach. We note that the resources needed to form classical shadows with bounded relative error depend strongly on the target state. We then comment on the advantages and limitations of using classical shadows to simulate many-body dynamics. In addition, we introduce the notion of a hybrid shadow, constructed from measurements on a part of the system instead of the entirety, which provides a framework to gain more insight into the nature of shadow states as one reduces the size of the subsystem measured, and a potential alternative to compressing quantum states.

Deep Reinforcement Learning for Preparation of Thermal and Prethermal Quantum States

We propose a method based on deep reinforcement learning that efficiently prepares a quantum many-body pure state in thermal or prethermal equilibrium. The main physical intuition underlying the method is that the information on the equilibrium states can be efficiently encoded/extracted by focusing on only a few local observables, relying on the typicality of equilibrium states. Instead of resorting to the expensive preparation protocol that adopts global features such as the quantum state fidelity, we show that the equilibrium states can be efficiently prepared only by learning the expectation values of local observables. We demonstrate our method by preparing two illustrative examples: Gibbs ensembles in non-integrable systems and generalized Gibbs ensembles in integrable systems. Pure states prepared solely from local observables are numerically shown to successfully encode the macroscopic properties of the equilibrium states. Furthermore, we find that the preparation errors, with respect to the system size, decay exponentially for Gibbs ensembles and polynomially for generalized Gibbs ensembles, which are in agreement with the finite-size fluctuation within thermodynamic ensembles. Our method paves a path toward studying the thermodynamic and statistical properties of quantum many-body systems in quantum hardware.

Exact Emergent Quantum State Designs from Quantum Chaotic Dynamics.

We present exact results on a novel kind of emergent random matrix universality that quantum many-body systems at infinite temperature can exhibit. Specifically, we consider an ensemble of pure states supported on a small subsystem, generated from projective measurements of the remainder of the system in a local basis. We rigorously show that the ensemble, derived for a class of quantum chaotic systems undergoing quench dynamics, approaches a universal form completely independent of system details: it becomes uniformly distributed in Hilbert space. This goes beyond the standard paradigm of quantum thermalization, which dictates that the subsystem relaxes to an ensemble of quantum states that reproduces the expectation values of local observables in a thermal mixed state. Our results imply more generally that the distribution of quantum states themselves becomes indistinguishable from those of uniformly random ones, i.e., the ensemble forms a quantum state design in the parlance of quantum information theory. Our work establishes bridges between quantum many-body physics, quantum information and random matrix theory, by showing that pseudorandom states can arise from isolated quantum dynamics, opening up new ways to design applications for quantum state tomography and benchmarking.

Quantum dynamics in the interacting Fibonacci chain

Quantum dynamics on quasiperiodic geometries has recently gathered significant attention in ultra-cold atom experiments where non trivial localised phases have been observed. One such quasiperiodic model is the so called Fibonacci model. In this tight-binding model, non-interacting particles are subject to on-site energies generated by a Fibonacci sequence. This is known to induce critical states, with a continuously varying dynamical exponent, leading to anomalous transport. In this work, we investigate whether anomalous diffusion present in the non-interacting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase. We investigate the dynamics of the interacting Fibonacci model by studying real-time spread of density-density correlations at infinite temperature using the dynamical typicality approach. We also corroborate our findings by calculating the participation entropy in configuration space and investigating the expectation value of local observables in the diagonal ensemble.

Gaussian continuous tensor network states for simple bosonic field theories

Tensor networks states allow one to find the low-energy states of local lattice Hamiltonians through variational optimization. Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally. However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically. In this paper we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic and quartic bosonic QFT Hamiltonians. We show that GCTNSs provide arbitrarily accurate approximations to the ground states of quadratic Hamiltonians and decent estimates for quartic ones at weak coupling. Since they capture the short distance behavior of the theories we consider exactly, GCTNSs even allow one to renormalize away simple divergences variationally. In the end our study makes it plausible that CTNSs are indeed a good manifold to approximate the low-energy states of QFTs.Received 18 December 2020Accepted 9 March 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023059Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum field theory (low energy)TechniquesTensor network methodsVariational approachQuantum Information

Eigenstate thermalization scaling in approaching the classical limit.

According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit-the number of sites and the particle number increasing at the same rate-the fluctuations should scale as ∼D^{-1/2} with the Hilbert space dimension D. Here, we study a different limit-the classical or semiclassical limit-by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.

Mathematically Proving Bell Nonlocality Motivated by the GHZ Argument

Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.

Extended nonergodic regime and spin subdiffusion in disordered SU(2)-symmetric Floquet systems

We explore thermalization and quantum dynamics in a one-dimensional disordered SU(2)-symmetric Floquet model, where a many-body localized phase is prohibited by the non-abelian symmetry. Despite the absence of localization, we find an extended nonergodic regime at strong disorder where the system exhibits nonthermal behaviors. In the strong disorder regime, the level spacing statistics exhibit neither a Wigner-Dyson nor a Poisson distribution, and the spectral form factor does not show a linear-in-time growth at early times characteristic of random matrix theory. The average entanglement entropy of the Floquet eigenstates is subthermal, although violating an area-law scaling with system sizes. We further compute the expectation value of local observables and find strong deviations from the eigenstate thermalization hypothesis. The infinite temperature spin autocorrelation function decays at long times as $t^{-\beta}$ with $\beta < 0.5$, indicating subdiffusive transport at strong disorders.

Predicting many properties of a quantum system from very few measurements

Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order $\log M$ measurements suffice to accurately predict $M$ different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

Integrable spin- Richardson–Gaudin XYZ models in an arbitrary magnetic field

We establish the most general class of spin- integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson–Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann–Feynman theorem.

From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

We derive exact formulas for the expectation value of local observables in a one-dimensional gas of bosons with point-wise repulsive interactions (Lieb–Liniger model). Starting from a recently conjectured expression for the expectation value of vertex operators in the sinh-Gordon field theory, we derive explicit analytic expressions for the one-point K-body correlation functions in the Lieb–Liniger gas, for arbitrary integer K. These are valid for all excited states in the thermodynamic limit, including thermal states, generalized Gibbs ensembles and non-equilibrium steady states arising in transport settings. Our formulas display several physically interesting applications: most prominently, they allow us to compute the full counting statistics for the particle-number fluctuations in a short interval. Furthermore, combining our findings with the recently introduced generalized hydrodynamics, we are able to study multi-point correlation functions at the Eulerian scale in non-homogeneous settings. Our results complement previous studies in the literature and provide a full solution to the problem of computing one-point functions in the Lieb–Liniger model.

Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations

Recent realization of a kinetically-constrained chain of Rydberg atoms by Bernien et al. [Nature 551, 579 (2017)] resulted in the observation of unusual revivals in the many-body quantum dynamics. In our previous work [arXiv:1711.03528] such dynamics was attributed to the existence of "quantum scarred" eigenstates in the many-body spectrum of the experimentally realized model. Here we present a detailed study of the eigenstate properties of the same model. We find that the majority of the eigenstates exhibit anomalous thermalization: the observable expectation values converge to their Gibbs ensemble values, but parametrically slower compared to the predictions of the eigenstate thermalization hypothesis (ETH). Amidst the thermalizing spectrum, we identify non-ergodic eigenstates that strongly violate the ETH, whose number grows polynomially with system size. Previously, the same eigenstates were identified via large overlaps with certain product states, and were used to explain the revivals observed in experiment. Here we find that these eigenstates, in addition to highly atypical expectation values of local observables, also exhibit sub-thermal entanglement entropy that scales logarithmically with the system size. Moreover, we identify an additional class of quantum scarred eigenstates, and discuss their manifestations in the dynamics starting from initial product states. We use forward scattering approximation to describe the structure and physical properties of quantum-scarred eigenstates. Finally, we discuss the stability of quantum scars to various perturbations. We observe that quantum scars remain robust when the introduced perturbation is compatible with the forward scattering approximation. In contrast, the perturbations which most efficiently destroy quantum scars also lead to the restoration of "canonical" thermalization.

Superluminal moving defects in the Ising spin chain

Quantum excitations in lattice systems always propagate at a finite maximum velocity. We probe this mechanism by considering a defect travelling at a constant velocity in the quantum Ising spin chain in transverse field. Independently of the microscopic details of the defect, we characterize the expectation value of local observables at large times and large distances from the impurity, where a Local Quasi Stationary State (LQSS) emerges. The LQSS is strongly affected by the defect velocity: for superluminal defects, it exhibits a growing region where translational invariance is spontaneously restored. We also analyze the behavior of the friction force exerted by the many-body system on the moving defect, which reflects the energy required by the LQSS formation. Exact results are provided in the two limits of extremely narrow and very smooth impurity. Possible extensions to more general free-fermion models and interacting systems are discussed

Stochastic locality and master-field simulations of very large lattices

In lattice QCD and other field theories with a mass gap, the field variables in distant regions of a physically large lattice are only weakly correlated. Accurate stochastic estimates of the expectation values of local observables may therefore be obtained from a single representative field. Such master-field simulations potentially allow very large lattices to be simulated, but require various conceptual and technical issues to be addressed. In this talk, an introduction to the subject is provided and some encouraging results of master-field simulations of the SU(3) gauge theory are reported.

Characterizing Time Irreversibility in Disordered Fermionic Systems by the Effect of Local Perturbations.

We study the effects of local perturbations on the dynamics of disordered fermionic systems in order to characterize time irreversibility. We focus on three different systems: the noninteracting Anderson and Aubry-André-Harper (AAH) models and the interacting spinless disordered t-V chain. First, we consider the effect on the full many-body wave functions by measuring the Loschmidt echo (LE). We show that in the extended or ergodic phase the LE decays exponentially fast with time, while in the localized phase the decay is algebraic. We demonstrate that the exponent of the decay of the LE in the localized phase diverges proportionally to the single-particle localization length as we approach the metal-insulator transition in the AAH model. Second, we probe different phases of disordered systems by studying the time expectation value of local observables evolved with two Hamiltonians that differ by a spatially local perturbation. Remarkably, we find that many-body localized systems could lose memory of the initial state in the long-time limit, in contrast to the noninteracting localized phase where some memory is always preserved.

Quasi locality of the GGE in interacting-to-free quenches in relativistic field theories

We study the quench dynamics in continuous relativistic quantum field theory, more specifically the locality properties of the large time stationary state. After a quantum quench in a one-dimensional integrable model, the expectation values of local observables are expected to relax to a generalized Gibbs ensemble (GGE), constructed out of the conserved charges of the model. Quenching to a free bosonic theory, it has been shown that the system indeed relaxes to a GGE described by the momentum mode occupation numbers. We first address the question whether the latter can be written directly in terms of local charges and we find that, in contrast to the lattice case, this is not possible in continuous field theories. We then investigate the less stringent requirement of the existence of a sequence of truncated local GGEs that converges to the correct steady state, in the sense of the expectation values of the local observables. While we show that such a sequence indeed exists, in order to unequivocally determine the so-defined GGE, we find that information about the expectation value of the recently discovered quasi-local charges is in the end necessary, the latter being the suitable generalization of the local charges while passing from the lattice to the continuum. Lastly, we study the locality properties of the GGE and show that the latter is completely determined by the knowledge of the expectation value of a countable set of suitably defined quasi-local charges.

The quench action approach in finite integrable spin chains

We consider the problem of constructing the stationary state following a quantum quench, using the exact overlaps for finite size integrable models. We focus on the isotropic Heisenberg spin chain with initial state Néel or Majumdar–Ghosh (dimer), although the proposed approach is valid for an arbitrary integrable model. We consider only eigenstates which do not contain zero-momentum strings because the latter are affected by fictitious singularities that are very difficult to take into account. We show that the fraction of eigenstates that do not contain zero-momentum strings is vanishing in the thermodynamic limit. Consequently, restricting to this part of the Hilbert space leads to vanishing expectation values of local observables. However, it is possible to reconstruct the asymptotic values by properly reweighting the expectations in the considered subspace, at the price of introducing finite-size corrections. We also develop a Monte Carlo sampling of the Hilbert space which allows us to study larger systems. We accurately reconstruct the expectation values of the conserved charges and the root distributions in the stationary state, which turn out to match the exact thermodynamic results. The proposed method can be implemented even in cases in which an analytic thermodynamic solution is not obtainable.

Generalized Gibbs ensemble in a nonintegrable system with an extensive number of local symmetries.

We numerically study the unitary time evolution of a nonintegrable model of hard-core bosons with an extensive number of local Z(2) symmetries. We find that the expectation values of local observables in the stationary state are described better by the generalized Gibbs ensemble (GGE) than by the canonical ensemble. We also find that the eigenstate thermalization hypothesis fails for the entire spectrum but holds true within each symmetry sector, which justifies the GGE. In contrast, if the model has only one global Z(2) symmetry or a size-independent number of local Z(2) symmetries, we find that the stationary state is described by the canonical ensemble. Thus, the GGE is necessary to describe the stationary state even in a nonintegrable system if it has an extensive number of local symmetries.