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Lieb--Robinson Bounds and Entanglement Limits in Quantum Dynamics with Finite-Dimensional Memory

Jeet Mozumdar

Published Apr 26, 2026• 0 citations
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Abstract

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 \[ \mathcal{H}_{\mathrm{ext}} = \bigotimes_{i=1}^{N} \mathcal{H}_i \otimes \bigotimes_{\text{bonds}} \mathcal{H}_M^{(i)}, \] 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 \(\tau\). More precisely, the memory channels satisfy \[ \left\| \mathcal{T}_M^s(X) - \mathrm{Tr}(X)\,\sigma_M \right\| \le c\, e^{-s/\tau}\, \|X\|, \] for all memory observables \(X\), where \(\sigma_M\) 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 \[ C_{i,j}(t) := \sup_{\substack{A_i = A_i^\dagger,\; B_j = B_j^\dagger \\ \|A_i\|\le 1,\; \|B_j\|\le 1}} \left| \mathrm{Tr}\!\left[(A_i \otimes B_j)\rho(t)\right] - \mathrm{Tr}\!\left[A_i \rho_i(t)\right]\, \mathrm{Tr}\!\left[B_j \rho_j(t)\right] \right|, \] satisfies an exponential bound of the form \[ C_{i,j}(t) \le C_0 \exp\!\Big[-\frac{d(i,j) - v_{\mathrm{eff}}(\tau)\, t}{\xi}\Big], \] where \(d(i,j)=|i-j|\) and \(\xi>0\) is independent of system size and time. The effective velocity \(v_{\mathrm{eff}}(\tau)\) 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 \(\tau \to \infty\), 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 \(\rho_{i,j}(t)\) and a maximally entangled state satisfies \[ F_{i,j}(t) = \langle \Phi^+ | \rho_{i,j}(t) | \Phi^+ \rangle \le \tfrac{1}{2}\bigl[1 + C_{i,j}(t)\bigr]. \] 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.

DOI
10.5281/zenodo.19842969

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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 [ \mathcal{H}{\mathrm{ext}} = \bigotimes{i=1}^{N} \mathcal{H}i \otimes \bigotimes{\text{bonds}} \mathcal{H}M^{(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 (\tau). More precisely, the memory channels satisfy [ \left| \mathcal{T}M^s(X) - \mathrm{Tr}(X),\sigma_M \right| \le c, e^{-s/\tau}, |X|, ] for all memory observables (X), where (\sigma_M) 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 [ 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|, ] satisfies an exponential bound of the form [ C_{i,j}(t) \le C_0 \exp!\Big[-\frac{d(i,j) - v_{\mathrm{eff}}(\tau), t}{\xi}\Big], ] where (d(i,j)=|i-j|) and (\xi>0) is independent of system size and time. The effective velocity (v_{\mathrm{eff}}(\tau)) 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 (\tau \to \infty), 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 (\rho_{i,j}(t)) and a maximally entangled state satisfies [ F_{i,j}(t) = \langle \Phi^+ | \rho_{i,j}(t) | \Phi^+ \rangle \le \tfrac{1}{2}\bigl[1 + C_{i,j}(t)\bigr]. ] 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.
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.
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.
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.
We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a “phase of matter.” Here, the combination of elementary Fourier analysis with the technique of Lieb–Robinson bounds (bounds on the velocity of propagation) is surprisingly powerful. We use this to prove exponential decay of connected correlation functions, a higher-dimensional Lieb–Schultz–Mattis theorem, and a Hall conductance quantization theorem for interacting electrons with disorder.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
The Lieb-Robinson bound shows that the speed of propagating information in a nonrelativistic quantum lattice system is bounded by a finite velocity, which entails the clustering of correlations. In this paper, we extend the Lieb-Robinson bound to quantum systems at finite temperature by calculating the dynamical correlation function at nonzero temperature for systems whose interactions are, respectively, short range, exponentially decaying, and long range. We introduce a simple way of counting the clusters in a cluster expansion by using the combinatoric generating functions of graphs. Limitations and possible applications of the obtained bound are also discussed.