Generalized Loschmidt echo and information scrambling in open systems

TL;DR

This work extends scrambling diagnostics to open quantum systems by generalizing the Loschmidt echo (LE) and the out-of-time-order correlator (OTOC) to Lindblad dynamics. Using a double-space (Choi–Jamiołkowski) framework, it analyzes LE dynamics under weak and strong dissipation, uncovering a universal single minimum in the weak regime and a two-minima structure in the strong regime when the Lindblad generator has a degenerate ground state. It further establishes open-system extensions of the OTOC–LE and OTOC–Rényi entropy relations and proposes an experimentally feasible protocol (e.g., with NMR) to measure OTOCs under dissipation. Collectively, the results offer a practical and theoretically grounded approach to diagnosing information scrambling in realistic, dissipative quantum platforms and suggest directions for exploring scrambling across a broader set of open-system measures.

Abstract

Quantum information scrambling, typically explored in closed quantum systems, describes the spread of initially localized information throughout a system and can be quantified by measures such as the Loschmidt echo (LE) and out-of-time-order correlator (OTOC). In this paper, we explore information scrambling in the presence of dissipation by generalizing the concepts of LE and OTOC to open quantum systems governed by Lindblad dynamics. We investigate the universal dynamics of the generalized LE across regimes of weak and strong dissipation. In the weak dissipation regime, we identify a universal structure, while in the strong dissipation regime, we observe a distinctive two-local-minima structure, which we interpret through an analysis of the Lindblad spectrum. Furthermore, we establish connections between the thermal averages of LE and OTOC and prove a general relation between OTOC and Rényi entropy in open systems. Finally, we propose an experimental protocol for measuring OTOC in open systems. These findings provide deeper insights into information scrambling under dissipation and pave the way for experimental studies in open quantum systems.

Figures (21)

• Figure 1: Schematic of the LE in closed (a) and open systems (b), as well as the OTOC in closed (c) and open systems (d). (a): In closed systems, the forward and backward time evolutions are governed by different Hamiltonians ($H_1$ and $H_2$), resulting in a final state (orange circle) that differs from the initial state (blue circle). (b): In open systems, due to interactions with the environment (Env.), the Lindbladians $\mathcal{L}_1$ and $\mathcal{L}_2$ govern the forward and backward time evolutions, respectively. (c-d): The time contours of the OTOC. In contrast to the time evolution of OTOC in closed systems (c), the forward and backward time evolution in open systems have correlations, as denoted by the orange curved lines in (d).
• Figure 2: The Loschmidt echo dynamics of the dissipative SYK model in the weak dissipation regime. We choose $N = 6$, $\gamma_1/J = 0.02$, and $\gamma_2/J = 0.1$, with dissipation applied to all Majorana fermions in the SYK model. The second Rényi entropy for the $\gamma_1$ and $\gamma_2$ evolutions are also plotted as blue and red dotted lines, respectively, for comparison. The initial state is the ground state of the SYK model. The time at which the Loschmidt echo reaches its minimum approximately coincides with the time when the second Rényi entropy for the smaller dissipation strength, $\gamma_1$, saturates to its final plateau value.
• Figure 3: The Loschmidt echo dynamics of the dissipative SYK model in the strong dissipation regime. Here, we choose $N=6$, $\gamma_1/J=10$, and $\gamma_2/J=100$, with the initial state being the ground state of the SYK model. In panel (a), dissipation is applied to all Majorana fermions, resulting in a non-degenerate ground state of $H_d$, while in panel (b), dissipation is applied to half of the Majorana fermions, leading to a degenerate ground state of $H_d$. The second Rényi entropy for $\gamma_1$ and $\gamma_2$ evolution are also plotted as the blue and red dot lines for comparison. The LE in (a) exhibits a single minimum structure, while the LE in (b) shows a two-local-minima structure.
• Figure 4: The schematic diagram illustrates the spectrum of $H_D$ in the strong dissipation regime, where the spectrum splits into segments along the imaginary axis, spaced by intervals on the order of $\gamma$. The ground state degeneracy of $H_D$ is indicated by the presence or absence of a border around the segment near zero, with this border having a width on the order of $J^2/\gamma$. The figure shows the case where the ground states of $H_D$ are degenerate, highlighted by the bordered segment near zero. In contrast, if the ground state is non-degenerate, the segment around zero appears without this border.
• Figure 5: The diagram represents the general operator $Q$ in (a) and the partial trace over subsystem B of it in (b).