In prior work by Mozumdar [1], a Lieb–Robinson-type bound for one-dimensional lat-tice systems evolving under discrete-time collision-model dynamics coupled to finite-dimensional,exponentially mixing bond memories was established.In this paper we see what survives when each bond memory is replaced by a separable, infinite-dimensional Hilbert space : the natural setting for physically realistic memory carriers such asbosonic modes, transmission-line resonators, or itinerant photonic pulses with a long but finitecoherence time. We show that the purely combinatorial, causal part of the finite-dimensionalargument (strict nearest-neighbor propagation, the light-cone counting bound) survives verbatim,since it never uses the dimension of the memory space. The mixing part of the argument doesnot survive verbatim: two structural facts used silently in the finite-dimensional proof : normequivalence on HM , and compactness of the unit ball of B(HM ) : fail in infinite dimensions, andboth are load-bearing. We introduce a weighted, energy-cutoff mixing hypothesis adapted from theopen-quantum-systems literature on Gaussian and quasi-free bosonic dynamics, show that it restoresa version of the memory-history contraction lemma, and sketch the resulting infinite-dimensionalLieb–Robinson bound. We flag precisely which steps are rigorous under the new hypothesis andwhich remain conjectural.
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), 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. More precisely, the memory channels satisfy
∥TMs(X)−Tr(X)σM∥≤ce−s/τ∥X∥, for all memory observables
X, where
σM is a stationary state, and
c>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):=supAi=Ai†,Bj=Bj†∥Ai∥≤1,∥Bj∥≤1∣Tr[(Ai⊗Bj)ρ(t)]−Tr[Aiρi(t)]Tr[Bjρj(t)]∣, satisfies an exponential bound of the form
Ci,j(t)≤C0exp[−ξd(i,j)−veff(τ)t], where
d(i,j)=∣i−j∣ and
ξ>0 is independent of system size and time. The effective velocity
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→∞, 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) and a maximally entangled state satisfies
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.
Memory effects in many-body quantum systems interacting with an environment significantly impact their behavior. Nevertheless, much of our understanding of many-body open quantum systems is based on models within the Born-Markov approximation where these effects are neglected. In this paper, we consider a general class of spatially local open quantum lattice models with a non-Markovian environment that is approximated as Gaussian. Our main result is a set of Lieb-Robinson bounds for such models. For models where the environment has independent local baths, we show that the Lieb-Robinson velocity is controlled by one-norm of the memory kernel within the evolution time-interval, yielding a linear light cone whenever the memory kernel has a bounded one-norm. For models where excitations can propagate through the environment, we prove a polynomial light cone that captures bath-mediated acceleration of information spreading. In both of these cases, we show that the derived Lieb-Robinson bounds are tight by also constructing lattice models saturating these bounds. Finally, we consider the problem of approximating a non-Markovian model by a larger Markovian model, which contains the system spins together with only a finite number of bosonic environment modes. We establish that as a consequence of our Lieb-Robinson bounds, the number of environment modes {per system site} needed to accurately capture local observables is independent of the size of the system.
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity.Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications.We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions.Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D → ∞.For example, in the Heisenberg model with spin S, we find v ≤ const.compared to the previous v ∝ S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v ∝ √ N instead of previous v ∝ N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model.Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models.Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.
The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time under a local Hamiltonian will essentially grow linearly in their support, up to exponentially suppressed corrections. In this work, we formulate Lieb-Robinson bounds for general harmonic systems on general lattices, for which the constituents are infinite-dimensional, as systems representing discrete versions of free fields or the harmonic approximation to the Bose-Hubbard model. We consider both local interactions as well as infinite-ranged interactions, showing how corrections to locality are inherited from the locality of the Hamiltonian: Local interactions result in stronger than exponentially suppressed corrections, while non-local algebraic interactions result in algebraic suppression. We derive bounds for canonical operators, Weyl operators and outline generalization to arbitrary operators. As an example, we discuss the Klein-Gordon field, and see how the approximate locality in the lattice model becomes the exact causality in the field limit. We discuss the applicability of these results to quenched lattice systems far from equilibrium, and the dynamics of quantum phase transitions.
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.