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arXiv:1003.3675 · Physical Review Letters

Lieb-Robinson Bound and Locality for General Markovian Quantum Dynamics

David Poulin

Published May 11, 2010• 138 citations
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Abstract

The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.

DOI
10.1103/physrevlett.104.190401

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The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.
We derive an extension of the Lieb--Robinson bound that accounts for memory effects in quantum lattice systems evolving under discrete-time non-Markovian dynamics generated by local interactions with finite-dimensional memory subsystems associated with each bond. The global evolution is unitary on an extended Hilbert space Hext=⨂i=1NHi⊗⨂bondsHM(i), \mathcal{H}_{\mathrm{ext}} = \bigotimes_{i=1}^{N} \mathcal{H}_i \otimes \bigotimes_{\text{bonds}} \mathcal{H}_M^{(i)}, Hext​=⨂i=1N​Hi​⊗⨂bonds​HM(i)​, while non-Markovian behavior, arising from repeated coupling to the same bond memory at successive time steps, leads to temporal correlations in the system. In this work, “finite memory” refers specifically to memory subsystems of fixed finite Hilbert space dimension whose reduced dynamics satisfy an exponential mixing condition in operator norm, meaning that correlation functions decay exponentially over time, with a characteristic timescale τ\tauτ. More precisely, the memory channels satisfy ∥TMs(X)−Tr(X) σM∥≤c e−s/τ ∥X∥, \left\| \mathcal{T}_M^s(X) - \mathrm{Tr}(X)\,\sigma_M \right\| \le c\, e^{-s/\tau}\, \|X\|, ∥TMs​(X)−Tr(X)σM​∥≤ce−s/τ∥X∥, for all memory observables XXX, where σM\sigma_MσM​ is a stationary state, and c>0c>0c>0 depends only on the memory dynamics. For a one-dimensional lattice with nearest-neighbor interactions, we prove that the connected correlation function, defined as Ci,j(t):=sup⁡Ai=Ai†,Bj=Bj†∥Ai∥≤1,∥Bj∥≤1∣Tr ⁣[(Ai⊗Bj)ρ(t)]−Tr ⁣[Aiρi(t)] Tr ⁣[Bjρj(t)]∣, C_{i,j}(t) := \sup_{\substack{A_i = A_i^\dagger,\; B_j = B_j^\dagger \\ \|A_i\|\le 1,\; \|B_j\|\le 1}} \left| \mathrm{Tr}\!\left[(A_i \otimes B_j)\rho(t)\right] - \mathrm{Tr}\!\left[A_i \rho_i(t)\right]\, \mathrm{Tr}\!\left[B_j \rho_j(t)\right] \right|, Ci,j​(t):=supAi​=Ai†​,Bj​=Bj†​∥Ai​∥≤1,∥Bj​∥≤1​​∣Tr[(Ai​⊗Bj​)ρ(t)]−Tr[Ai​ρi​(t)]Tr[Bj​ρj​(t)]∣, satisfies an exponential bound of the form Ci,j(t)≤C0exp⁡ ⁣[−d(i,j)−veff(τ) tξ], C_{i,j}(t) \le C_0 \exp\!\Big[-\frac{d(i,j) - v_{\mathrm{eff}}(\tau)\, t}{\xi}\Big], Ci,j​(t)≤C0​exp[−ξd(i,j)−veff​(τ)t​], where d(i,j)=∣i−j∣d(i,j)=|i-j|d(i,j)=∣i−j∣ and ξ>0\xi>0ξ>0 is independent of system size and time. The effective velocity veff(τ)v_{\mathrm{eff}}(\tau)veff​(τ) is explicitly bounded in terms of the local interaction strength and the memory mixing parameters, ensuring the system exhibits a finite light cone, meaning that information propagation is confined to a region growing linearly with time, with a well-defined maximum speed. In the limit τ→∞\tau \to \inftyτ→∞, the bound no longer guarantees a finite propagation speed, reflecting the transition to a persistent memory regime where correlations do not decay exponentially. As an operational consequence, the entanglement fidelity between two-site reduced states ρi,j(t)\rho_{i,j}(t)ρi,j​(t) and a maximally entangled state satisfies Fi,j(t)=⟨Φ+∣ρi,j(t)∣Φ+⟩≤12[1+Ci,j(t)]. F_{i,j}(t) = \langle \Phi^+ | \rho_{i,j}(t) | \Phi^+ \rangle \le \tfrac{1}{2}\bigl[1 + C_{i,j}(t)\bigr]. Fi,j​(t)=⟨Φ+∣ρi,j​(t)∣Φ+⟩≤21​[1+Ci,j​(t)]. These results extend Lieb--Robinson theory to a class of discrete-time quantum dynamics with memory subsystems associated with each bond, enabling resummation of multi-step memory effects into an effective correlation propagation kernel that governs the decay of temporal correlations. This establishes a quantitative relationship between the timescale of memory mixing and the effective speed of information propagation in the system.
The Lieb-Robinson bound (LRB) states that the range and strength of interactions between the constituents of a complex many-body system impose upper limits to how fast the signal can propagate. It manifests in a light conelike growth of correlation function connecting two distant subsystems. Here, we employ the techniques of quantum information to demonstrate that the LRB can be determined from local measurements performed on a single qubit that is connected to a many-body system. This formulation provides an operational recipe for estimating the LRB in complex systems, replacing the measurement of the correlation function with simple single-particle manipulations. We demonstrate the potency of this approach by deriving the upper limit to the speed of signal propagation in the XY spin chain.
We study the velocity of the propagation of information for a class of local dissipative quantum dynamics. This finite velocity is expressed by the so-called Lieb-Robinson bound. Besides the properties of the already studied dynamics, we consider an additional relation that expresses the propagation of certain subspaces. The previously derived bounds did not reflect the dissipative character of the dynamics and yielded the same result as for the reversible case. In this article, we show that for this class the velocity of propagation of information is time dependent and decays in time towards a smaller velocity. In some cases the velocity becomes zero. At the end of the article, the exponential clustering theorem of general frustration free local Markovian dynamics is revisited.
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).
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.
The speed of light ccc sets a strict upper bound on the speed of information transfer in both classical and quantum systems. In nonrelativistic quantum systems, the Lieb-Robinson Theorem imposes an emergent speed limit v≪cv \hspace{-0.2mm} \ll \hspace{-0.2mm} cv≪c, establishing locality under unitary evolution and constraining the time needed to perform useful quantum tasks. We extend the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the expectation that measurements can arbitrarily violate spatial locality, we find at most an (M+1)(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)(M+1)-fold enhancement to the speed vvv of quantum information, provided the outcomes of measurements in MMM local regions are known. This holds even when classical communication is instantaneous, and extends beyond projective measurements to weak measurements and other nonunitary channels. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols. We tightly constrain the resource requirements for quantum computation, error correction, teleportation, and generating entangled resource states (Bell, GHZ, quantum-critical, Dicke, W, and spin-squeezed states) from short-range-entangled initial states. Our results impose limits on the use of measurements and active feedback to speed up quantum information processing, resolve fundamental questions about the nature of measurements in quantum dynamics, and constrain the scalability of a wide range of proposed quantum technologies.