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 $|\psi ⟩$, ${\mathcal{O}}_x(t){\mathcal{O}}_y|\psi ⟩\approx {\mathcal{O}}_y{\mathcal{O}}_x(t)|\psi ⟩$ with arbitrarily small errors—so long as $|x-y|\gtrsim vt$, 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 $V(r)\propto 1/r^{\alpha }$. 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 $|\psi ⟩$ when $\alpha >3$ (Lieb-Robinson light cone); for a typical state $|\psi ⟩$ chosen uniformly at random from the Hilbert space when $\alpha >\frac{5}{2}$ (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.

Popular 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.

Figure 1

The hierarchy of linear light cones in one dimension; we say that a light cone has exponent $\gamma$ if $\parallel [A_0(t),B_r]\parallel$ is large only when $t\gtrsim r^{\gamma }$. The plot depicts the exponents of the Lieb-Robinson light cone (solid line) [21], the Frobenius light cone from Theorem 7 (dot-dashed line), and the free light cone from Theorem 9 (dashed line) as functions of $\alpha$ in one dimension. The free light cone is known to be a tight bound for all $\alpha$. We also show that the Lieb-Robinson and Frobenius light cones are not linear below $\alpha =3$ and $\alpha =\frac{5}{2}$, respectively.

Figure 2

The norm of the interaction between two balls ${\mathcal{B}}_{1,2}$ of size $t$, separated by a distance $r$, determines the shape of the light cone. The critical values of $\alpha$ after which this norm becomes large differ depending on whether we use the operator or Frobenius norm and whether the system is interacting or free.

Figure 3

An illustration of our single-particle state-transfer protocol in $d=2$ dimensions. Through four steps, we redistribute a particle initially at 0 to the square $B_1^{(0)}$, then to $B_2^{(0)}$ and $B_3^{(0)}$, and finally to $B_4^{(0)}$, each time doubling the size of the region sharing the particle. We use different colors to mark the additional sites that the particle spreads to in different steps. We then reverse the process to concentrate the particle on the target site $x$ and thereby achieve a perfect state transfer. The protocol is enhanced by the volume $r^d$ of the squares, with $r$ being a typical size of the squares. This enhancement offsets the penalty $1/r^{\alpha }$ due to the power-law constraint, resulting in a superlinear state-transfer protocol when $\alpha <d+1$.

Figure 4

A protocol for rapid growth of the commutator norm using two-body long-range interactions. Step 1: We use cnot gates between nearest-neighbor sites to spread a single Pauli $X_0$ to a Pauli string $XX\dots X$ supported on every site inside a ball of radius $O(t)$ centered at $X_0$. Step 2: We use pairwise $ZZ$ interactions between all sites in the two balls, located distance $O(r)$ apart, which adds an operator of norm $\sim O(t^{2d+1}/r^{\alpha })$ into the second ball a distance $r$ away. Step 3: We invert step 1 in the outer ball, pushing all of the operator weight in the outer ball onto a single site.

Figure 5

The $4^L$-dimensional space of operators can be decomposed into the direct sum of $L$ subspaces $\{{\mathbb{Q}}_i\}$ by the position of the rightmost occupied site. By keeping track of only the “average value” of the rightmost site (depicted above), keeping in mind that an exponential number of orthogonal operators (depicted below) are contained on most of the sites, we reduce the quantum walk of many-body operators from an exponentially large space to a one-dimensional line.

Figure 6

A depiction of the initial state in Ref. [10]. Empty circles represent empty lattice sites, and filled circles represent initially occupied sites.