Hierarchy of Linear Light Cones with Long-Range Interactions

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 vertical bar psi >, O-x(t)O-y vertical bar psi > approximate to OyOx(t)vertical bar psi > with arbitrarily small errors-so long as vertical bar x -y vertical bar greater than or similar to vt, where v is finite. Yet, most nonrelativistic physical systems realized in nature have long-range interactions: Two degrees of freedom separated by a distance r interact with potential energy V(r) proportional to 1/r(alpha). In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: At the same a, 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 vertical bar psi > when alpha > 3 (Lieb-Robinson light cone); for a typical state vertical bar psi > chosen uniformly at random from the Hilbert space when alpha > 5/2 (Frobenius light cone); and for every state of a noninteracting 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, is bounded by the Frobenius light cone and, therefore, is poorly constrained by all Lieb-Robinson bounds.