Judging by the enormous body of work that it has inspired, Elliott Lieb and Derek Robinson’s 1972 article on the “Finite group velocity of quantum spin systems” can be regarded as a high-impact paper, as research accountants say. But for more than 30 years, this major contribution to quantum physics has remained pretty much confidential. Lieb and Robinson’s work eventually found a large audience in the years 2000, with the rapid and concomitant development of quantum information theory and new experimental platforms. In this chapter, I will first remind the reader of the central result of Lieb and Robinson’s work, which is the emergence of a local causality structure in the dynamics of non-relativistic quantum systems, and manifests by the exponential suppression of the commutator of any two operators outside an effective light cone in space-time. I will then review the experiments that most closely relate to this finding, in the sense that they reveal the group velocity of information propagation in “real” quantum systems. Finally, as an outlook, I will attempt to make a connection with the quantum version of the butterfly effect recently studied in chaotic quantum systems.
Judging by the enormous body of work that it has inspired, Elliott Lieb and Derek Robinson’s 1972 article on the “Finite group velocity of quantum spin systems” can be regarded as a high-impact paper, as research accountants say. But for more than 30 years, this major contribution to quantum physics has remained pretty much confidential. Lieb and Robinson’s work eventually found a large audience in the years 2000, with the rapid and concomitant development of quantum information theory and new experimental platforms. In this chapter, I will first remind the reader of the central result of Lieb and Robinson’s work, which is the emergence of a local causality structure in the dynamics of non-relativistic quantum systems, and manifests by the exponential suppression of the commutator of any two operators outside an effective light cone in space-time. I will then review the experiments that most closely relate to this finding, in the sense that they reveal the group velocity of information propagation in “real” quantum systems. Finally, as an outlook, I will attempt to make a connection with the quantum version of the butterfly effect recently studied in chaotic quantum systems.
Judging by the enormous body of work that it has inspired, Elliott Lieb and Derek Robinson's 1972 article on the "Finite Group Velocity of Quantum Spin Systems" can be regarded as a \emph{high-impact paper}, as research accountants say. But for more than 30 years this major contribution to quantum physics has remained pretty much confidential. Lieb and Robinson's work eventually found a large audience in the years 2000, with the rapid and concomitant development of quantum information theory and experimental platforms enabling the characterisation and manipulation of isolated quantum systems at the single-particle level. In this short review article, I will first remind the reader of the central result of Lieb and Robinson's work, namely the existence of a maximum group velocity for the propagation of information in non-relativistic quantum systems. I will then review the experiments that most closely relate to this finding, in the sense that they reveal how information propagates in specific -- yet "real" -- quantum systems. Finally, as an outlook, I will attempt to make a connection with the quantum version of the butterfly effect recently studied in chaotic quantum systems.
A re there limits on how fast a physical effect can propagate in a system? The answer to this fundamental question has far-reaching consequences for practically any dynamic system we want to use or understand, from media for sending long-distance messages to extended bodies approaching thermal equilibrium. In a pioneering 1972 study, Elliott Lieb and Derek Robinson showed that the speed at which a physical effect could travel through a collection of interacting particles always has an upper limit [1]. The existence of the Lieb-Robinson bound, however, assumes that the particles are locally interacting, whereas many forms of "complex matter" of interest today involve long-range interactions. In a trio of papers, researchers now establish a bound for many long-range systems as well [2-4]. The findings settle a recent debate over if and when such bounds exist, and they may have implications for the choice of platform for quantum computation and other quantum technologies. You may have raised an eyebrow upon reading that the Lieb-Robinson bound was considered a discovery. After all, special relativity says that the speed of light is the ultimate limit on how fast a signal can travel, and this limit is built into the quantum field theory that describes the interactions between elementary particles. But researchers are also interested in whether an intrinsic limit on the speed of a physical effect exists simply by virtue of the way a system's particles interact. This limit determines, for instance, how quickly a jolt to an atom in a solid will influence a distant atom's behavior. When Lieb and Robinson explored this bound, they used a lattice of spins as a generic model of matter on a microscopic scale, where quantum mechanics prevails. The interactions in their model refer, for instance, to the energy cost of two spins pointing in opposite directions. The interactions are also short-ranged because the energy cost falls off exponentially with the distance between two spins, as exp(−x/d). Lieb and Robinson defined their "physical effect" as a correlation between two distant spins. Consider, for instance, a lattice with all spins initially aligned upwards (Fig. 1). Now imagine flipping one spin at x = 0. It will mainly stir its closest neighbors, which will stir theirs, and so on, and it will take some time t before a spin at a distant point x feels the initial flip. Lieb and Robinson quantified this correlation onset between distant spins with a so-called commutator, a mathematical object involving the quantum spin operator acting at x = 0 and t = 0 and the quantum spin operator acting at x and t. They showed that there is always a velocity v such that the "norm" of this commutator is negligibly small when x > vt. In modern language, physicists would say the speed of a physical effect is bound to an effective light cone (Fig. 1)-akin to the causal light cone of relativity. The slope of this cone is v, and the fact that it's constant means that the bound doesn't speed up or slow down with time.
As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound-the Lieb-Robinson bound-and the butterfly effect in strongly coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity v_{B}. Similarly, the Lieb-Robinson velocity places a state-independent ballistic upper bound on the size of time evolved operators in nonrelativistic lattice models. Here, we argue that v_{B} is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free-particle computations to understand the role of strong coupling. We find that v_{B} remains constant or decreases with decreasing temperature. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.
The Lieb-Robinson bound (LRB) states that the range and strength of interactions between the constituents of a complex many-body system impose upper limits to how fast the signal can propagate. It manifests in a light conelike growth of correlation function connecting two distant subsystems. Here, we employ the techniques of quantum information to demonstrate that the LRB can be determined from local measurements performed on a single qubit that is connected to a many-body system. This formulation provides an operational recipe for estimating the LRB in complex systems, replacing the measurement of the correlation function with simple single-particle manipulations. We demonstrate the potency of this approach by deriving the upper limit to the speed of signal propagation in the XY spin chain.
This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
The butterfly velocity was recently proposed as a characteristic velocity of chaos propagation in a local system. Compared to the Lieb-Robinson velocity that bounds the propagation speed of all perturbations, the butterfly velocity, studied in thermal ensembles, is an "effective" Lieb-Robinson velocity for a subspace of the Hilbert space defined by the microcanonical ensemble. In this paper, we generalize the concept of butterfly velocity beyond the thermal case to a large class of other subspaces. Based on holographic duality, we consider the code subspace of low energy excitations on a classical background geometry. Using local reconstruction of bulk operators, we prove a general relation between the boundary butterfly velocities (of different operators) and the bulk causal structure. Our result has implications in both directions of the bulk-boundary correspondence. Starting from a boundary theory with a given Lieb-Robinson velocity, our result determines an upper bound of the bulk light cone starting from a given point. Starting from a bulk space-time geometry, the butterfly velocity can be explicitly calculated for all operators that are the local reconstructions of bulk local operators. If the bulk geometry satisfies Einstein equation and the null energy condition, for rotation symmetric geometries we prove that infrared operators always have a slower butterfly velocity that the ultraviolet one. For asymptotic AdS geometries, this also implies that the butterfly velocities of all operators are upper bounded by the speed of light. We further prove that the butterfly velocity is equal to the speed of light if the causal wedge of the boundary region coincides with its entanglement wedge. Finally, we discuss the implication of our result to geometries that are not asymptotically AdS, and in particular, obtain constraints that must be satisfied by a dual theory of flat space gravity.