Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

We thank the referee for their careful consideration of our work and detailed comments. Here we respond to their requested changes directly and outline how we have improved the manuscript accordingly.

Point 1:

"The first is about the claimed impossibility to synchronize two ½ spins. The authors argument stands on (their) unusual definition of synchronization for which two temporal signals:
\begin{equation*}
\cos(t) \quad \& \quad a + \cos (t)
\end{equation*}
would not be synchronized, due to the presence of an offset.
Generally synchronization is considered looking at the dynamical part of signals and any constant off-set is actually neglected. Beyond the authors choice of “terminology”, they should notice that actually these signals would be perfectly synchronized also using the same synchronization indicator of their Eqs. 1 and 2."

Firstly, we must emphasise that we have nowhere claimed that it is impossible synchronize two spin-1/2s. In our response to the referee's previous comments, we provided an example of two spin-1/2s under a common Markovian bath and explained that under our stricter definition, these two spins do not anti-synchronize but that \emph{``under a weaker definition of synchronization we could consider this system to be (anti-)synchronized''.

We chose to work with the strictest definition of synchronization in our work to avoid additional technicalities in the main theorems as these technicalities offer little additional insight. In our previous revision we already made indication of where additional considerations could be made to obtain weaker notions of synchronization. In our latest version we have included a new subsection dedicated to this discussion in order to make these ideas clearer. We have also explicitly pointed out to the reader that our definitions are stricter that the Pearson correlation indicator.

Point 2:
"The authors answer:
“The claim that we cannot treat subsystems that are not identical is incorrect. We now give an explicit example in Sec. 6.2.1 ...”
Maybe there was a misunderstanding, as I was referring specifically to synchronization.
Indeed “stable quantum synchronization” is addressed in Section 3.2 providing sufficient conditions in two corollaries. Both of them assume the exchange/permutation symmetry between sites, the first in the Liouvillian and the second in the operator A, rather strong symmetries indeed at the basis of my comment.
The authors do not refer to this in their answer but highlight the example in Sec. 6.2.1, actually already in the previous version of the manuscript. This is a rather elaborated model and in the present form it does not clarify this point.
It is clear that the global $S^+$ and $S^-$ are permutation invariant, while the Hamiltonian is not.
On the other hand, in order to appreciate the claimed generality, could the authors specify the local operators $O_j$ that would be identically synchronized in this inhomogeneous model?
It would be also useful to include the definition of $c_j$ in the “agnostic” local ladders (38) and (39)."

We now recognise our misunderstanding, and apologise to the referee. Our framework can only treat systems where the local Hilbert spaces of each site have the same dimension - i.e. we cannot consider synchronizing a spin-1/2 with a spin-1. We have made this clearer in our revised manuscript, and explicitly pointed out that our definition for robustness would no longer be applicable for sites with different local Hilbert space dimensions. We have also outlined in our conclusions that extending our theory for studying synchronisation between subsystems with different local Hilbert spaces is an active focus of future work.

We now recognise our misunderstanding, and apologise to the referee. Our framework can only treat systems where the local Hilbert spaces of each site have the same dimension - i.e. we cannot consider synchronizing a spin-1/2 with a spin-1. We have made this clearer in our revised manuscript, and explicitly pointed out that our definition for robustness would no longer be applicable for sites with different local Hilbert space dimensions. We have also outlined in our conclusions that extending our theory for studying synchronisation between subsystems with different local Hilbert spaces is an active focus of future work.

Finally, we thank the referee for identifying a typographical mistake, Eqs. (38) and (38) should read
\begin{align*}
L^{-}_{j}&=\gamma^-_{j} c_{j, \uparrow}c_{j, \downarrow}\\
L^+_{j}&=\gamma^+_{j} c^\dagger_{j, \uparrow}c^\dagger_{j, \downarrow}
\end{align*}

Point 3:
"As a note, the newly added theorem 4 would be really less obscure clarifying the meaning of the “commutant” proportional to the Identity."

This has been clarified by a footnote.

We thank the referee for their careful consideration of our work and detailed comments. Here we respond to their requested changes directly and outline how we have improved the manuscript accordingly.

Point 1:
"The authors refined their introduction to the Lindblad equation. Nevertheless, I would appreciate if the authors decided for a uniform approach as to whether they focus on the Markovian "spirit" of the master equation or whether they want to emphasise non-Markovian extensions. The authors argue that one may place part of the environment inside the system's Hilbert space to allow for non-Markovian dynamics. However, one might argue that the actual environment represented by the Lindblad operators is still purely Markovian. Note that there also exist alternative extensions to bring the Lindblad formalism to the non-Markovian regime, see e.g. the work by Breuer (Phys. Rev. A 75, 022103) or the work by Head-Marsden and Maziotti (Phys. Rev. A 99, 022109), but in my view these are all extensions to the - in itself Markovian - Lindblad formalism. Although my concern is primarily semantic, I think it is nonetheless worthwhile to stick as close as possible to the established literature when it comes to discussing the Lindblad master equation framework.
In any case, the discussion in Appendix K is much appreciated. Nevertheless, I think it would be important to include the central conclusion "[...] it is not possible to draw general conclusions directly about the dynamics of the system without knowing the details of the non-Markovian bath and applying our results to the combined system." in the main text."

As advised by the referee, we have now restricted the discussion in the main text so that it focuses exclusively on the Markovian regime. We have then amended our appendix slightly and included the references suggested by the referee.

Point 2:
"Finally, I would recommend another thorough proofreading of the text to amend the remaining grammatical issues. For example, in the introduction it should be "characterize" in the sentence "These results provide a general theory for studying quantum synchronization and characterizes the phenomenon." Another example is at the end of the introductory text for 4.1 where I suppose that by "We also remark that generically there is only one order of s different between the decay rate and the oscillation frequency." the authors probably mean "[...] the order of s only differs by one between the decay rate [...]", i.e., it is not about the existence of a different order but rather that the order itself is different, if I understand it well."

We thank the referee for their careful reading of our manuscript. As advised, we have thoroughly proof read the manuscript again and eliminated the remaining grammatical issues found.