Generalized Loschmidt echo and information scrambling in open systems

1- I believe the paper addresses and interesting problem: the generalisation of the concept of fidelity, Loschmidt echo and OTOC to open systems

2- The paper outlines clearly the parallel between the definitions proposed for open systems and the accepted ones for thermally isolated ones.

3- The paper is clearly written.

1- Though the authors discuss in detail the connection between the definitions presented and those for closed systems the physical meaning of the quantities is hardly discussed in the open system case.

2- The example discussed is in my opinion too simplistic and far from being universal

1- I believe the manuscript lacks a discussion of the physical meaning of the proposed quantities. As stated after Eq.(1) for closed systems the "LE measures the sensitivity of quantum evolution to the perturbation and quantifies the degree of irreversibility". What about open systems ? It seems to me that the physics is slightly different: closed systems do not posses a globale stationary state (only reduced density matrices may have one) while in this case there is one and its dependence on the perturbation distinguishing L_1 and L_2 determines the asymptotic value of M^D(t) . If it is one than there is no sensitivity of the stationary state to a change of system parameters. How should one interpret the intermediate dynamics ? Same question for the OTOC: the physical meaning of the OTOC for closed systems is connected to the square commutator and its semiclassical representation suggesting that its exponential growth will detect many body chaos. What kind of information would one get from the proposed quantity ? What motivates physically its usefulness for open systems ?

2- Are the generalisations of the LE and OTOC to open systems unique ? A discussion of this point would be useful.

3- In view of point 1 of this section the example chosen is quite peculiar: the stationary state is insensitive to the system parameters, hence the LE always tends to one. Since at t=0 starts at one, it cannot do much but have one or more local minima which are described in the manuscript but whose physical meaning is not discussed. What happens if one considers a more interesting situation in which the stationary state does depend on the system parameters and on dissipation ?

Ask for minor revision

1. The paper addresses a timely and important problem—the study of scrambling and irreversibility in open quantum systems.
2. The authors introduce a generalized version of the Loschmidt Echo (LE), designed to capture irreversible dynamics in open systems, and investigate its behavior numerically within the SYK model.
3. They explore several intriguing properties of their new quantity, generalizing known results to the open setting scenario: 3.1) A generalization of the known relationship between the LE and out-of-time-order correlators (OTOCs) is developed for open systems. 3.2) The authors discuss a connection between the generalized LE and the so-called Renyi entropies in open quantum systems. 3.3) A potential experimental implementation of the generalized LE is proposed.
4. Overall, the paper is clear, well-structured, and well written

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.

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.

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.

4. While the second Renyi 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.

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.

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