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arXiv:0808.3581 · Physical Review Letters

Supersonic Quantum Communication

Jens Eisert, D. Gross

Published Jun 17, 2009• 57 citations
arXivHTMLPDF

Abstract

When locally exciting a quantum lattice model, the excitation will propagate through the lattice. This effect is responsible for a wealth of nonequilibrium phenomena, and has been exploited to transmit quantum information. It is a commonly expressed belief that for local Hamiltonians, any such propagation happens at a finite "speed of sound". Indeed, the Lieb-Robinson theorem states that in spin models, all effects caused by a perturbation are essentially limited to a causal cone. We show that for meaningful translationally invariant bosonic models with nearest-neighbor interactions (addressing the challenging aspect of an experimental realization) this belief is incorrect: We prove that one can encounter accelerating excitations under the natural dynamics that allow for reliable transmission of information faster than any finite speed of sound. It also implies that the simulation of dynamics of strongly correlated bosonic models may be much harder than that of spin chains even in the low-energy sector.

DOI
10.1103/physrevlett.102.240501

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When locally exciting a quantum lattice model, the excitation will propagate through the lattice. This effect is responsible for a wealth of nonequilibrium phenomena, and has been exploited to transmit quantum information. It is a commonly expressed belief that for local Hamiltonians, any such propagation happens at a finite "speed of sound". Indeed, the Lieb-Robinson theorem states that in spin models, all effects caused by a perturbation are essentially limited to a causal cone. We show that for meaningful translationally invariant bosonic models with nearest-neighbor interactions (addressing the challenging aspect of an experimental realization) this belief is incorrect: We prove that one can encounter accelerating excitations under the natural dynamics that allow for reliable transmission of information faster than any finite speed of sound. It also implies that the simulation of dynamics of strongly correlated bosonic models may be much harder than that of spin chains even in the low-energy sector.
On physical grounds, one expects locally interacting quantum many-body systems to feature a finite group velocity. This intuition is rigorously underpinned by Lieb-Robinson bounds that state that locally interacting Hamiltonians with finite-dimensional constituents on suitably regular lattices always exhibit such a finite group velocity. This also implies that causality is always respected by the dynamics of quantum lattice models. It had been a long-standing open question whether interacting bosonic systems also feature finite speeds of sound in information and particle propagation, which was only recently resolved. This work proves a strikingly simple such bound for particle propagation - shown in literally a few elementary, yet not straightforward, lines - for generalized Bose-Hubbard models defined on general lattices, proving that appropriately locally perturbed stationary states feature a finite speed of sound in particle numbers.
We study the nonequilibrium dynamics of correlations in quantum lattice models in the presence of long-range interactions decaying asymptotically as a power law. For exponents larger than the lattice dimensionality, a Lieb-Robinson-type bound effectively restricts the spreading of correlations to a causal region, but allows supersonic propagation. We show that this decay is not only sufficient but also necessary. Using tools of quantum metrology, for any exponents smaller than the lattice dimension, we construct Hamiltonians giving rise to quantum channels with capacities not restricted to a causal region. An analytical analysis of long-range Ising models illustrates the disappearance of the causal region and the creation of correlations becoming distance independent. Numerical results obtained using matrix product state methods for the XXZ spin chain reveal the presence of a sound cone for large exponents and supersonic propagation for small ones. In all models we analyzed, the fast spreading of correlations follows a power law, but not the exponential increase of the long-range Lieb-Robinson bound.
The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/r^{α} at distance r? Here, we present a definitive answer to this question for all exponents α>2d and all spatial dimensions d. Schematically, information takes time at least r^{min{1,α-2d}} to propagate a distance r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.
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 ρ ∝ exp½-μN (with N the total boson number), ensemble-averaged correlators of the form h½A 0 ; B r ðtÞi, along with outof-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 ½A 0 ; B r ðtÞ 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 highdensity 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.
In this work, we investigate how quickly local perturbations propagate in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, these systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on a specific but experimentally natural situation in which the number of bosons at any one site in the unperturbed initial state is approximately limited. We rigorously prove the existence of an almost-linear information-propagation light cone, thus establishing a Lieb-Robinson bound: the wave front grows at most as t log^{2}(t). We prove the clustering theorem for gapped ground states and study the time complexity of classically simulating one-dimensional quench dynamics, a topic of great practical interest.
We study the speed at which information propagates through systems of interacting quantum particles moving on a regular lattice and show that for a certain class of initial conditions there exists a maximum speed of sound at which information can propagate. Our argument applies equally to quantum spins, bosons such as in the Bose-Hubbard model, fermions, anyons, and general mixtures thereof, on arbitrary lattices of any dimension. It also pertains to dissipative dynamics on the lattice, and generalizes to the continuum for quantum fields. Our result can be seen as an analog of the Lieb-Robinson bound for strongly correlated models.