Independent Quantum Science and Engineering Researcher
Quantum Information Theory | Mathematical & Many-Body Methods in Quantum Systems | Quantum Networks, Channels & Entanglement.
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:
\[ C_{i,j}(t) := \sup_{\substack{A_i = A_i^\dagger,\; B_j = B_j^\dagger \\ \|A_i\|\le 1,\; \|B_j\|\le 1}} \left| \mathrm{Tr}\!\left[(A_i \otimes B_j)\rho(t)\right] - \mathrm{Tr}\!\left[A_i \rho_i(t)\right]\, \mathrm{Tr}\!\left[B_j \rho_j(t)\right] \right| \]
satisfies the exponential bound:
\[ C_{i,j}(t) \le C_0 \exp\!\Big[-\frac{d(i,j) - v_{\mathrm{eff}}(\tau)\, t}{\xi}\Big] \]
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 that 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.
Download PDF Lieb–Robinson Preprint
@article{mozumdar2026,
author = {Jeet Mozumdar},
title = {Lieb--Robinson Bounds and Entanglement Limits in Quantum Dynamics with Finite-Dimensional Memory},
year = {2026},
doi = {10.5281/zenodo.19842970},
publisher = {Zenodo}
}
Last updated: 26 April 2026