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arXiv:quant-ph/0603121 · Physical Review Letters

Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order

Sergey B. Bravyi, M. B. Hastings, Frank Verstraete

Published Jul 31, 2006• 662 citations
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Abstract

The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block.

DOI
10.1103/physrevlett.97.050401

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The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block.
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.
The dynamics of quantum systems strongly depends on the local structure of the Hamiltonian. For short-range interacting systems, the well-known Lieb–Robinson bound defines the effective light cone with an exponentially small error with respect to the spatial distance, whereas we can obtain only polynomially small errors for distance in long-range interacting systems. In this paper, we derive a qualitatively new bound for quantum dynamics by considering how many spins can correlate with each other after time evolution. Our bound characterizes the number of spins which support the many-body entanglement with exponentially small errors and is valid for a large class of Hamiltonians including long-range interacting systems. To demonstrate the advantage of our approach in quantum many-body systems, we apply our bound to prove several fundamental properties which have not been derived from the Lieb–Robinson bound.
We consider the variation of von Neumann entropy of subsystem reduced states of general many-body lattice spin systems due to local quantum quenches. We obtain Lieb-Robinson-like bounds that are independent of the subsystem volume. The main assumptions are that the Hamiltonian satisfies a Lieb-Robinson bound and that the volume of spheres on the lattice grows at most exponentially with their radius. More specifically, the bound exponentially increases with time but exponentially decreases with the distance between the subsystem and the region where the quench takes place. The fact that the bound is independent of the subsystem volume leads to stronger constraints (than previously known) on the propagation of information throughout many-body systems. In particular, it shows that bipartite entanglement satisfies an effective ``light cone,'' regardless of system size. Further implications to ttt density-matrix renormalization-group simulations of quantum spin chains and limitations to the propagation of information are discussed.
In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.
Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp  ( v t − r ) , where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on r d sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp  ( − r d ) . We confirm this intuition, showing that, for r > v t , the volume-filling operator is suppressed by exp  ( − ( r − v t ) d / ( v t ) d − 1 ) . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t : as ε becomes sufficiently small, only ε − O ( t d − 1 ) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.
The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound.We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1=r α .The bound implies an effective light cone tighter than all previous bounds.Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18.To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish.This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation.In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions.In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3D (where D is the number of dimensions).