A Lieb-Robinson bound for open quantum systems with memory

TL;DR

This work proves a Lieb-Robinson bound for spatially local non-Markovian open quantum lattice models with Gaussian baths, showing that finite environment memory time yields a linear light cone for system dynamics, with velocity $\,v_{LR} = e a_0 \mathcal{Z} + 56 e a_0 \mathcal{Z}\, TV(\text{U})$. It develops an operator-space framework using Wick contractions to handle unbounded environments and derives quasi-locality bounds for the system-channel dynamics. The authors then establish that non-Markovian dynamics can be well-approximated by a larger Markovian dilation that couples the system to a finite number of environment modes per site, with the required modes per site being independent of system size and depending only on evolution time and desired precision. They also present a concrete memory-rich counterexample showing that infinite memory (infinite $TV(K)$) can violate the linear light cone, clarifying the bound's scope. Collectively, the results provide rigorous tools for simulating non-Markovian open quantum many-body systems and understanding information propagation under memory effects.

Abstract

We consider a general class of spatially local non-Markovian open quantum lattice models, with a bosonic environment that is approximated as Gaussian. Under the assumption of a finite environment memory time, formalized as a finite total variation of the memory kernel, we show that these models satisfy a Lieb-Robinson bound. Our work generalizes Lieb Robinson bounds for open quantum systems, which have previously only been established in the Markovian limit. Using these bounds, we then show that these non-Markovian models can be well approximated by a larger Markovian model, which contains the system spins together with only a finite number of environment modes. In particular, 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.

Figures (6)

• Figure 1: Schematic depiction of the non-Markovian many-body model considered in this paper. The many-body system is defined on a $d-$dimensional lattice, $\Lambda$, and interacts locally to a non-Markovian bath via the system operators $R^{\nu}_\alpha$ and memory kernels $\textnormal{K}^{\nu, \nu'}_\alpha$.
• Figure 2: Schematic depicting a time-dependent 1D model with supersonic transport that violates a Lieb-Robinson bound with a linear light cone. The model is constructed by interleaving three different time-dependent system-environment Hamiltonians --- for $0 \leq t \leq 2T$, the system qubits are continuous excited and they consequently transfer a total of $T$ particles into the bath. Then, from $2T \leq t \leq 2 T + 1$, the first qubit in the system is excited and from $2T + 1 \leq t \leq 3T + 1$, the bath oscillators, which have $T$ particles in them, are used to mediate a transport of this excitation at a velocity $\sim \sqrt{T}$.
• Figure 3: Schematic depiction of the star-to-chain transformation used to approximate a non-Markovian environment by a discrete set of bosonic modes. Each bath is replaced with $N_m$ modes, with $N_m$ controls the accuracy of this approximation, and these modes themselves 1D nearest-neighbour coupled lattice.
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Theorems & Definitions (55)

• Proposition 1
• Proposition 2
• Definition 1: Operator space $\mathcal{S}(\rho_E)$
• Definition 2: Operator space $\mathcal{Q}(\rho_E)$
• Lemma 1: Wick's theorem
• Lemma 2
• proof
• Definition 3: Function space $\text{PC}^{1}_{t^*}(\mathbb{R})$
• Definition 4
• Lemma 3