Finite speed of quantum information in models of interacting bosons at finite density

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 density matrix $\rho \propto \exp[-\mu N]$ (with $N$ the total boson number), ensemble averaged correlators of the form $\langle [A_0,B_r(t)]\rangle$, along with out-of-time-ordered correlators, must vanish as the distance $r$ between two local operators grows, unless $t \ge r/v$ 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 $[A_0,B_r(t)]$ 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.