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 density matrix ρ ∝ 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.

Maximal speed for macroscopic particle transport in the Bose-Hubbard model

“…Our result is a new kind of macroscopic-type Lieb-Robinson bound for particle transport. It complements other recent results [35][36][37][38] which hold for special initial states and are otherwise closer to the original formulation of the Lieb-Robinson bound.…”

“…This condition excludes states of positive local density, e.g., Mott states (9). Very recently, a number of groups have made progress on this problem through novel techniques: The N -scaling of the velocity was improved to √ N [35]; an almost-linear light cone was derived for special initial states that are local perturbations of a stationary state satisfying certain exponential constraints on the local particle density [36]; a linear light cone was derived for commutators tested against the state e −µN [37]; and [34] was extended to propagation through vacuum [38].…”

Exponential clustering of bipartite quantum entanglement at arbitrary temperatures

“…In such systems, we do not usually obtain the Lieb-Robinson bound with a finite Lieb-Robinson velocity [180]. To extend our results, we may have to restrict ourselves to particular classes of quantum many-body boson systems, such as free boson systems [181,182], spin-boson models [183,184], and Bose-Hubbard type Hamiltonians [185][186][187]. The establishment of the Lieb-Robinson bound in the boson systems is still an active area of research.…”

Bounds in Nonequilibrium Quantum Dynamics

“…( 82) and ( 87) with respect to some low-density initial states have been proved for a class of Bose-Hubbard-like model. 144,145,146…”