Abstract
We prove that quantum information propagates with a finite velocity in any model of interacting bosons whose (possibly time-dependent) Hamiltonian contains spatially local single-boson hopping terms along with arbitrary local density-dependent interactions. More precisely, with the density matrix (with the total boson number), ensemble-averaged correlators of the form , along with out-of-time-ordered correlators, must vanish as the distance between two local operators grows, unless for some finite speed . In one-dimensional models, we give a useful extension of this result that demonstrates the smallness of all matrix elements of the commutator between finite-density states if is sufficiently small. Our bounds are relevant for physically realistic initial conditions in experimentally realized models of interacting bosons. In particular, we prove that can scale no faster than linear in number density in the Bose-Hubbard model: This scaling matches previous results in the high-density limit. The quantum-walk formalism underlying our proof provides an alternative method for bounding quantum dynamics in models with unbounded operators and infinite-dimensional Hilbert spaces, where Lieb-Robinson bounds have been notoriously challenging to prove.
- Received 25 June 2021
- Accepted 22 March 2022
DOI:https://doi.org/10.1103/PhysRevX.12.021039
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In a nonrelativistic model of quantum spins, with each spin placed on grid points in a lattice and interacting only with its neighbors, there is an emergent and finite speed at which quantum information can be sent. This result, proved in a now-classic theorem in 1972, is highly reminiscent of Einstein’s prior observation that information cannot propagate faster than light in relativity. Experiments and simulations have suggested that interacting bosons also have emergent speed limits, but a rigorous proof has remained elusive. Here, we prove that in experimentally realizable models of interacting Bose gases with density-dependent interactions, there is a finite speed of information.
Our proof applies to all states with a finite boson density in one dimension and to typical finite density states in higher dimensions. It also applies to driven and undriven systems. In one dimension, our result implies that while Bose gases are not harder to simulate than conventional spin models, they also cannot enhance the speed of quantum information processors relative to those built out of spins or fermions.
Our result reveals fundamental limits on the speed of quantum information processing in experimentally realizable settings, and it sharpens our understanding of the dynamics of quantum information and thermalization in Bose gases realized in both condensed matter and atomic physics.