A Speed Test for Ripples in a Quantum System

Abstract

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.