Generalized Loschmidt echo and information scrambling in open systems

We thank Referee B for the positive assessment of our work and for the valuable comments and questions, which have helped us improve the manuscript. Because some of the equations in our responses are not compatible with the Markdown editor of this website, we have included our detailed replies in the attached PDF file.

Attachment:

replyB.pdf

Reply to Referee A

We thank Referee A for recognizing the importance and clarity of our work and for viewing it as opening a promising research direction. We also appreciate the referee’s helpful comments regarding the generality of our claims and the assumptions behind our analytical results. In response, we have clarified these assumptions throughout the manuscript and added numerical results for a quantum spin model to further support our conclusions. Detailed responses are provided below.

Referee A:
1. In discussing LE dynamics in open systems, the authors focus exclusively on the SYK model, despite claiming that this choice comes "without loss of generality". While the SYK model is indeed a well-established example of quantum chaos, it possesses several specific features—such as disorder and all-to-all interactions—that may not generalize to more realistic many-body systems. This raises open questions about the role of locality, conservation laws, and other system-specific properties. I suggest the authors temper the generality of their claims and clarify—particularly in Section 3, as well as in the introduction and conclusions—that their results are derived within the SYK model. Alternatively, they could support their claims by demonstrating similar behavior in a different, perhaps more physically grounded, many-body system.

Reply:
We thank the referee for this helpful suggestion. Although the SYK model is a well-known example of a strongly interacting quantum system, we agree that relying on it alone is not sufficient to support the universality of our conclusions. To further substantiate our claims, we now present additional numerical results for a local spin-$1/2$ model, demonstrating that the dynamical behavior of the open-system LE is generic and not restricted to the SYK model. Specifically, we consider the dissipative XXZ spin chain with Hamiltonian

We then subject the XXZ chain to local dissipation generated by $L_j = S^z_j$ on each site. Our numerical simulations for this model show the same qualitative LE behavior as in the SYK case: a single minimum in the weak-dissipation regime and a two–local-minima structure in the strong-dissipation regime.

More concretely, for the parameters shown in Fig. 17 (weak dissipation, $\Delta/J = 2$, $\gamma_1/J = 0.02$, $\gamma_2/J = 0.1$), the open-system LE exhibits a single minimum, in direct analogy with the SYK results. In the strong-dissipation regime [Fig. 16, $\Delta/J = 2$, $\gamma_1/J = 10$, $\gamma_2/J = 100$], the LE instead displays a clear two-minima structure, again analogous to the SYK case with a degenerate ground space of $H_d$. We observe the same pattern in other phases of the XXZ chain: the gapped ferromagnetic phase at $\Delta/J = -1.5$ [Figs. 18, 19] and the gapless Luttinger-liquid phase at $\Delta/J = 0.5$ [Figs. 20, 21]. In all these cases, weak dissipation leads to a single minimum in the LE, while strong dissipation produces two local minima.

These results support our claim that the LE behavior we discuss is not specific to the SYK model. In the weak-dissipation regime, the single minimum follows directly from a perturbative expansion in the dissipation strength and does not rely on any special property of the SYK Hamiltonian. In the strong-dissipation regime, the two-minima structure can be traced to the segmented structure of the Lindblad spectrum: the Hamiltonian acts as a perturbation to the dissipative generator, and when it does not commute with the dissipative part, it lifts degeneracies in the latter’s eigenstates. This generates a separation of time scales, with short-time dynamics controlled by the $\gamma t$ scale and long-time dynamics by the $J^2 t/\gamma$ scale, which in turn produces two distinct minima in the LE.

In summary, the emergence of two local minima in the open-system LE only requires that (i) the Hermitian part of the doubled-space Hamiltonian does not commute with the dissipative part, and (ii) the ground space of the dissipative part in doubled space is degenerate (as is the case, for example, for generic local and translation-invariant Lindblad operators). These conditions are not special to the SYK model and are expected to hold for a broad class of open quantum systems.

We have added the XXZ simulations and a discussion of these points to Appendix C. In addition, we have revised the main text (at the end of Sec. 3, in the introduction, and by adding a second paragraph to the conclusion section) to clarify that, although our explicit calculations are performed in the SYK and XXZ models, the mechanism underlying the LE behavior is general.

Referee A:
2. In Eq. (20), the term $V_{\alpha'}$ appears. While its origin can be inferred from the diagrammatic discussion in the appendix, a clearer explanation of how Eq. (20) is derived would be helpful, particularly in Section 4.1.

Reply:
$V_{\alpha'}$ and $V_{\alpha}$ are independent noise operators acting on subsystem $B$. In Eq. (20), the average OTOC involves two copies of $B(t)$, and under the noise-averaged approximation, the noise on each copy is independent. We denote these independent noise operators as $V_{\alpha}$ and $V_{\alpha'}$. In Eq. (20), ${V_{\alpha}}$ denotes the noise operators and the overline indicates averaging over their realizations. Here we use random external noise on subsystem $B$ to approximate the role of the interaction between the subsystem and its environment, a standard approach in studies of decoherence dynamics. This approximation can be easily understood in the $\delta \ll 1$ limit: the time evolution of an operator $R_B$ can be expanded to order $\delta^2$, leading to an effective master equation under the Born–Markov approximation. This master equation is equivalent to a stochastic time evolution of subsystem $B$ under a Langevin noise ${V_\alpha}$. To clarify this point, we have added a sentence in Section 4.1 below Eq. (19) explaining the noise-averaged approximation, and we have inserted an additional step in Eq. (20) to make the derivation more transparent.

Referee A:
3. The derivations in Sections 4 and 5 are based on averaging over unitary operators, which implies that the established relations hold for the ensemble-generalized LE and "averaged" OTOC. Additionally, the connection to Renyi entropies applies only to a specific class of operators, namely density matrices. It would strengthen the paper if the authors were more explicit about these assumptions and limitations—ideally in both the introduction and the conclusions.

Reply:
We thank Referee A for this valuable suggestion. To improve the clarity and rigor of the manuscript, we have revised both the introduction and the conclusions to explicitly state these assumptions. We have also clarified that the OTOC–Rényi entropy relation is derived for a broad class of density operators and the corresponding OTOCs.

Referee A:
4. While the second Rényi entropy is a well-known measure of entanglement for pure states in closed quantum systems, its interpretation in open quantum systems is less clear. It would be valuable if the authors could elaborate on its operational meaning or relevance in this context.

Reply:
We thank the Referee for this constructive suggestion. We have added a paragraph in our revised manuscript to clarify the physical meaning of the second Rényi entropy in open systems. In brief, the second Rényi entropy $S_2$ quantifies information lost from the system to its environment. For a pure state under closed Hamiltonian evolution, $S_2 = 0$, whereas coupling to a bath or performing measurements increases $S_2$ through decoherence and dissipation. In a Lindblad description, its growth is set by the decoherence rates, and $S_2(t)$ can be used to extract dephasing rates, mixing times, and the Liouvillian gap. For subsystems in many-body open dynamics, the growth and saturation of $S_2$ capture the competition between unitary entanglement generation and monitoring/dissipation and serve as diagnostics of dissipative or measurement-induced phase transitions. Moreover, $S_2$ is directly measurable via SWAP or randomized-measurement protocols on two copies of the system.

We have incorporated this discussion to better motivate our generalization to open systems, and have added one paragraph at the end of Sec. 5 of the main text.

Referee A:
5. The protocol proposed in Section 5 relies on the ability to implement time-reversed Lindblad dynamics. While this is known to be feasible in closed systems—such as in NMR experiments by inverting the Hamiltonian sign—it remains unclear how such a reversal could be achieved in open systems. A more detailed discussion of possible implementations or theoretical frameworks for time-reversed Lindblad evolution would enhance the practical relevance of the proposed protocol.

Reply:
We thank the Referee for this helpful suggestion. The measurement protocol in Sec. 6, which concerns the experimental procedure for measuring the OTOC, indeed requires a time-reversal step. In closed systems, this can be implemented by flipping the sign of the Hamiltonian, as demonstrated experimentally (e.g., Jun Li et al., Phys. Rev. X 7, 031011 (2017)). In our open-system setting governed by a Lindblad equation, the corresponding “time reversal” is implemented by the conjugate Lindbladian evolution defined in Eq. (23). For the Hermitian jump operators ($L_k = L_k^\dagger$) considered in our work, a direct comparison of Eqs. (22) and (23) shows that the only change is $H \rightarrow -H$, while the dissipator remains invariant. Thus, the experimental implementation is identical to the closed-system case: one simply reverses the sign of the Hamiltonian while keeping the dissipative part unchanged. We have added this clarification in the final paragraph of Sec. 6 of the revised manuscript.