Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach

1- New methods for deriving Lindblad for given steady state
2-Method is potentially simple

1-Examples are solely for benchmarking the theory

The authors Leonardo da Silva Souza and Fernando Iemini derive in their draft with the title ''Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach'' a method to calculate the coefficients scaling predefined Hamiltonian terms and jump operators that evolve a quantum system into a given stationary state. The method is quite general and assumes time evolution of density operators with Lindblad master equations. The authors demonstrate that finding the coefficients require to solve a linear system of L x K^2 dimension where L is the number of Hamiltonian terms and K the number of jump operators. They apply this approach to harmonic oscillators where they predict how to generate coherent and squeezed states and to a collective spin model.

My personal opinion is that this draft is very interesting and the method that the authors apply can be useful for many different disciplines working with open quantum systems. With this I argue that this draft is potentially suitable for SciPost Physics. The biggest weakness of this draft are the examples and before I come to several minor criticims I would like the authors to address this major point.

Major: I would have expected that the authors use their developed method to derive new results but for some reason the authors only discuss examples which, I would say, only benchmark their theory. While this is definitely a good check, I would appreciate if the authors could add one new example. Alternatively, I would also be happy if the authors could at least clarify what are the new insights that one gains from these known examples.

Minor:

(i) This is maybe because of my ignorance: it took me a while to understand that the steady state but also the hermitian and jump operators are given. Could the authors clarify this before Eq. (4), where you write the steady state is known. Maybe just add there that also the hermitian and jump operators are know. I guess in an experiment this would mean you have a certain but possibly not full control over the system. Maybe the authors could comment?

(ii) The authors write: ''This function computes roughly the norm [...]'' Why roughly?

(iii) I believe the challenge in the authors' method, which is basically captured by Eq.(9), is the calculation of the elements of the matrix M. This would in general require to calculate complicated operator products. Can the authors comment on this? How does one calculate M for a more general situation where one cannot rely on analytical results as in the examples.

(iv) I did not understand the comment on ''to explore non-Markovian maps''. What does this approach have to do with non-markovian maps? Why is that relevant and how does it connect to the non-positivity of gamma?

(v) I find the comment on ''unique possible solution in Lindblad form'' a bit missleading because there are in my opinion many solutions as soon as one adds a frequency of the harmonic oscillator (h3=a^dag a). Maybe the authors could add a comment.

(vi) The authors write ''Squeezed states could be generated [...]''. I think the authors should write ''can'' instead.

(vii) For a harmonic oscillator with linear driving, dissipation does in general stabilize the state even when driven on resonance. For a parameterically driven oscillator this is not necessarily the case (especially when driven on parametric resonance which the authors study because they have not added the a^dag a term for h). I guess this is what the authors rediscover here (because they find these two possibly unphysical solutions in Eqs.(17)). Is it the case that the two solutions describe an amplification without proper steady state? Also what happens if one adds a finite detuning?

(viii) Related to (vii). Is the motivation of introducing two-particle dissipation that single particle dissipation is insufficient to stabilize the parametric resonance?

(ix) Can the authors remind the reader when they write about Strong dissipative (and weak dissipative) case what they mean by this? Please just repeat \kappa/\omega_0>(<)0.

(x) The authors write ''On the other hand, due to the vanishing of the smallest eigenvalue with system size it suggests that the method should still be feasible for large system sizes''. My question is where this implication comes from? I mean, for instance, is the infinite temperature state (Eq.(23)) for N to infinity even a proper limit?

Ask for minor revision

1. Originality: The proposed method offers a new approach to reconstructing Lindblad master equations. The method's scalability, with quadratic dependence on the number of jump operators, is a significant improvement compared to methods with exponential growth.

2. Potential Applications: The method's applicability to a wide range of quantum systems, including those with large Hilbert spaces, suggests its potential for practical use in various fields.

3. Clear Presentation: The manuscript is well-written and presents the concepts and the mathematical methods clearly and understandably.

1. Limited Scope of Examples: The focus on systems with macroscopic parameters limits the generalizability of the method. A broader range of examples would demonstrate its applicability to more complex systems.

2. Lack of Comparative Analysis: While the authors stress the computational efficiency of their method, a more detailed comparison to existing techniques would make the argument more solid

The authors present a method to reconstruct the form of a Lindblad master equation by reverse engineering using non-equilibrium steady states. Examples of the method based on Gaussian bosonic and collective spin systems are presented. The main result is that the complexity of the problem scales with the square of the number of jump operators, thus avoiding exponential growth with the size of the Hilbert space. The authors also show that their approach can be used as a no-go theorem to detect whether a Lindbladian has a steady state by looking at the absence of null eigenvalues of the correlation matrix (instead of the full Lindbladian).

Comments:

From a theoretical point of view, the manuscript looks very interesting as it offers a clear advantage over some brute-force methods, but the computational advantage over other existing techniques is not discussed in any detail.

The two examples discussed describe the case where the dynamics of the dissipative system can be described by some macroscopic parameters. However, it is not clear how difficult this would be for a general case.

The authors promise to discuss cases beyond the Markovian limit for the master equation. However, the discussion of how the method deals with non-Markovian maps is somewhat lacking.

Minor comment: the word "Lindbladian" is sometimes spelled "Lindbaldian".

In conclusion, I think that the manuscript meets the criteria for acceptance by SciPost Physics. I am favorable toward the publication in SciPost Physics once the authors have adequately addressed the comments presented in my report.

1. Provide a more quantitative comparison of the computational complexity of the proposed method with other existing techniques.
2. Include at least the recipe to be followed in an example that does not have simplifying macroscopic parameters.
3. Discuss the challenges and limitations of applying the method to non-Markovian systems, or diminish what is promised in the manuscript.

Ask for minor revision