Finite Speed of Quantum Information in Models of Interacting Bosons at Finite Density

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 ?exp[-µN] (with N the total boson number), ensemble-averaged correlators of the form «[A0,Br(t)]», along with out-of-time-ordered correlators, must vanish as the distance r between two local operators grows, unless t=r/v for some finite speed v. In one-dimensional models, we give a useful extension of this result that demonstrates the smallness of all matrix elements of the commutator [A0,Br(t)] between finite-density states if t/r is sufficiently small. Our bounds are relevant for physically realistic initial conditions in experimentally realized models of interacting bosons. In particular, we prove that v 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.