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, $\mathcal{O}_x(t) \mathcal{O}_y |\psi\rangle \approx \mathcal{O}_y\mathcal{O}_x(t) |\psi\rangle$ with arbitrarily small errors -- so long as $|x-y|\gtrsim vt$, where $v$ is finite. Yet most non-relativistic physical systems realized in nature have long-range interactions: two degrees of freedom separated by a distance $r$ interact with potential energy $V(r) \propto 1/r^{\alpha}$. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: at the same $\alpha$, 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 when $\alpha>3$ (Lieb-Robinson light cone); for a typical state chosen uniformly at random from the Hilbert space when $\alpha>\frac{5}{2}$ (Frobenius light cone); for every state of a non-interacting system when $\alpha>2$ (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, are bounded by the Frobenius light cone, and therefore are poorly constrained by all Lieb-Robinson bounds.

Highlights

  • While nonrelativistic quantum systems do not possess intrinsic absolute speed limits, their dynamics exhibit a form of causality analogous to the speed of light

  • Theorem 11 proves that any possible improvement to Theorem 9 must be subalgebraic. Both the linear light cone and the superlinear polynomial light cones we prove for free quantum systems with long-range interactions are known to be optimal

  • We have demonstrated a hierarchy of linear light cones—

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Summary

Introduction

While nonrelativistic quantum systems do not possess intrinsic absolute speed limits, their dynamics exhibit a form of causality analogous to the speed of light. Lieb and Robinson first deduced the existence of a finite velocity for the propagation of information in quantum spin systems with finite-range interactions [1] This velocity leads to ballistic dynamics, out of which a linear light cone emerges. For systems with power-law interactions, i.e., those that fall off as 1=rα in the distance r between two degrees of freedom, the story is much richer Such long-range interactions are exhibited in a variety of quantum simulators and technological platforms, including ultracold atomic gases [2], Rydberg atoms [3], one-dimensional chains of trapped ions [4], polar molecules [5], color centers in solid-state systems [6], and atoms trapped in photonic crystals [7]. In addition to being interesting from this fundamental-science perspective, 2160-3308=20=10(3)=031009(29)

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