Hierarchy of Linear Light Cones with Long-Range Interactions

Abstract

In quantum many-body systems with local interactions, quantum information and entanglement cannot spread outside of a linear light cone, which expands at an emergent velocity analogous to the speed of light. Local operations at sufficiently separated spacetime points approximately commute—given a many-body state|ψ⟩,Ox(t)Oy|ψ⟩≈OyOx(t)|ψ⟩with arbitrarily small errors—so long as|x−y|≳vt, wherevis finite. Yet, most nonrelativistic physical systems realized in nature have long-range interactions: Two degrees of freedom separated by a distancerinteract with potential energyV(r)∝1/rα. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: At the sameα, some quantum information processing tasks are constrained by a linear light cone, while others are not. In one spatial dimension, this linear light cone exists for every many-body state|ψ⟩whenα>3(Lieb-Robinson light cone); for a typical state|ψ⟩chosen uniformly at random from the Hilbert space whenα>52(Frobenius light cone); and for every state of a noninteracting system whenα>2(free light cone). These bounds apply to time-dependent systems and are optimal up to subalgebraic improvements. Our theorems regarding the Lieb-Robinson and free light cones—and their tightness—also generalize to arbitrary dimensions. We discuss the implications of our bounds on the growth of connected correlators and of topological order, the clustering of correlations in gapped systems, and the digital simulation of systems with long-range interactions. In addition, we show that universal quantum state transfer, as well as many-body quantum chaos, is bounded by the Frobenius light cone and, therefore, is poorly constrained by all Lieb-Robinson bounds.

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Plain Language Summary

Long-range interactions in quantum systems can be leveraged to boost the performance of many quantum technologies such as computing, simulation, and metrology. Until recently, it was not yet clear if such systems impose a fundamental limit on the rate at which information can be transferred, analogous to the speed of light in relativistic systems. But recent work has shown that such a limit does exist in quantum systems whose long-range interactions decay as a power law with distance. Here, we take this a step further and mathematically demonstrate a hierarchy of such speed limits, or “light cones”: a series of increasingly stringent bounds on increasingly specific kinds of quantum information spreading.

Our work is a breakthrough in constraining and engineering quantum dynamics to be as useful as possible. The hierarchy of light cones reveals that the standard mathematical framework for quantum information dynamics does not effectively constrain practical tasks that include quantum state transfer, nor do standard techniques effectively constrain quantum chaos and thermalization. Our explicit protocols show the tightness of the hierarchy of bounds and also how these long-ranged systems can spread information faster than local systems.

These results demonstrate the ultimate limits of quantum information transfer in quantum technologies that exploit long-ranged interactions such as trapped ions, cold gases of polar molecules, or atoms in photonic crystals.