Dissipative flow equations

1
Timely, interesting issue
2
Advocating a novel approach to Lindbladians, transferred from
Hamiltonians.

1
Renormalizing property of generators not discussed properly

2
Interpretation of eigenvalues of Lindbladian not complete
in terms of physical significance.

3
Incomplete representation of literature

4
Some weaknesses of presentation (line widths, misleading wording, upper/lower case letters)

The authors examine whether some standard infinitesimal
generators of unitary flows, which were introduced for simplifying Hamiltonians and their dynamics, can also be used for Lindbladians. Given the rising interest in dealing with open quantum systems, the issue is interesting and timely. Althoug the approach is only tested for special, solvable cases the
submitted manuscript gets a relevant message across and deserves publication.

There are some points which can still be improved to increase the
impact of the paper. They are given below:

1)
End of page 1, beginning of page 2: In the list of physics issues for which flow equations have been used, quantum magnetism, high order perturbation theory and bound states ought to be mentioned as well, e.g., by citing
Knetter et al., EPJB 13, 209 (2000); PRL 87, 167204 (2001)
Windt et al. PRL 87, 127002 (2001);
Powalski et al. SciPost 4, 1 (2018)

2)
The remark after Eq. (16) is misleading as it suggests that there is a fundamental difference between the approach for Hamiltonians and for Lindbladians. But for both of them, one can either transform all operators in the new basis or transform H or L, respectively, forward and backward. Hence there is no conceptual difference between these two applications.

3)
At several instances, the authors state that generator 3 is optimum because of its uniform convergence. But they also state that generator 2 is advantageous if truncations are necessary. This is not a side remark, but a rather crucial point:
If the method is used *without* any approximation the precise choice of generator is only a matter of convenience. Then, a fast convergence is certainly desirable for numerical implementation. If, however, the *renormalizing* property is crucial because certain terms are truncated, the matrix elements
linking states of very different eigen energies should better be transformed first. Hence, generator 1 and 2 have this conceptual asset while generator 3 does not. This should be discussed and stated
clearly.

4)
Some graphs are difficult to understand because the lines are to thin and the colors too weak due to dotted lines, e.g., Fig. 4 and Fig. 7.

5)
When eigenvalues of the Lindbladian are discussed, for instance, in Sect. 5,
their imaginary part occurs with both signs. Clearly, one sign corresponds
to exponential decay, the other to exponential growth. In an open system, however, only the decay makes sense. Hence, the question is urging why the "wrong" sign occurs and how it happens that it does not matter in physics. The authors should clarify this very important point.

6)
Related to 5), I find it most helpful it the temporal evolution
of a physical quantity, for instance an occupation, is computed
and displayed so that one sees the convergence towards the
stationary states.

7)
What is the physics of a "strongly dissipative state"?
Can you give a definition for such a state? Which signature
would an experimentalist have to look for?

8)
In the discussion of the different generators, the authors use the
term "decay" to describe the convergence towards diagonality. I find
this wording misleading since in disspative system a true physical decay
routinely occurs - but this is physics and not a property of the
mathematical basis change. Please formulate the mathematical property
a bit clearer, i.e., less ambiguously.

Minor points are:

a)
The word "modelisation" in the very first paragraph exists in French,
but not in English as far as I know.

b)
The use of lower and upper case letters in the reference got screwed - please
improve it, e.g., bec -> BEC and so on.

Add some additional discussions to remove weaknesses 1 and 2,
add references and improve working where appropriate, see report.

I have read the paper “Dissipative flow equations” by Lorenzo Rosso et al. where a theory generalising flow equations, first introduced for Hamiltonian systems, to open quantum dynamics is presented.
I find the paper really interesting and generally well written. The method presented is also nicely put into context and well motivated. Beyond presenting the general theory the authors discuss several examples which are of interest and of help to understand the theoretical results.

I have only one concern with the paper and this is about the notation used in section 2.
The notation does not make clear whether the transformation $\mathcal{S}$, and thus also its generator $\eta$, is a matrix of the same dimension of $\rho$ or whether it is actually a super-operator as is the Lindblad super-operator, i.e. a map acting on matrices such as $\rho$.

I try to explain myself.
In Eq.(2) $\mathcal{S}$ seems to have the same role of a Lindblad super-operator and thus one would conclude that $\mathcal{S}$ is a map acting over matrices. As such, also $\eta$ in Eq.(3) would be a similar object.
The commutators in Eq.(4) would thus be commutators between maps, i.e. the difference that one has when acting first with a map and then with the other or the other way around.
However, in Eq.(9-10) it actually seems that $\mathcal{S}$ is a matrix just like $\rho$ or $O$, even more when looking at Eq.(12), which suggests that the generator of $\mathcal{S}$, $\eta$, is a matrix like $O$. This is also the case for the last equation in pag. 5.

Also Eq.(14) seems to treat $\mathcal{S}$ as a simple matrix (by the way I think that the $\rho(t)$ inside the square bracket in the definition of $\rho_\ell (t)$ should not have a time-dependence).
However, in the line just below Eq.(14), one finds the relation $\mathcal{S}(\ell)e^{\mathcal{L}t}\mathcal{S}^{-1}(\ell)=e^{\mathcal{L}(\ell)t}$ which seems to suggest again that $\mathcal{S}$ is a super-operator as is the Lindblad, otherwise I don't see how $\mathcal{S}$ and $\mathcal{S}^{-1}$ can be brought at the exponent. However, in Eq.(16) $\eta$, which must be an object similar to $\mathcal{S}$, seems instead a matrix again.

If I look at the examples where the Lindblad is basically always represented in a doubled space, I get the feeling that $\mathcal{S}$ should indeed be a super-operator like the Lindblad. In this representation, they are both matrices but the density matrix is instead a vector.

I fear that my confusion with the notation emerges because the authors do not make clear distinction between super-operators and matrices.
One way could be to always write that the super-operator acts on a matrix. For instance, in this notation, if $\mathcal{S}$ is really a super-operator, Eq.(2) would read as

$\mathcal{L}(\ell)[X]=\mathcal{S}(\ell)\circ \mathcal{L}\circ \mathcal{S}(\ell)^{-1}[X]$

Where $\circ$ denotes composition of superoperators and $X$ indicates that the relation holds for any matrix $X$. The above equation clearly defines $\mathcal{L}(\ell)$ through its action on any $X$, as the map obtained by acting on X first with the super-operator $\mathcal{S}^{-1}(\ell)$, then with the super-operator $\mathcal{L}$ and finally with $\mathcal{S}(\ell)$. Of course, all other equations should be adjusted accordingly.
If it is true that $\mathcal{S}$ is a super-operator then I do not understand the meaning of equations 9-10 where $\mathcal{S}$ is treated as a matrix. If $\mathcal{S}$ is a superoperator I would expect Eq.9 to read as
$$
{\rm Tr}\left(\mathcal{S}^{-1}(\ell)\circ\mathcal{S}(\ell)[\rho]O\right)
$$
where the action of $\mathcal{S}^{-1}$ can then be passed to act onto $O$. I would also expect Eq. (10) to be something like
$$
\rho_0(\ell)=\mathcal{S}(\ell)[\rho]
$$

It seems like the authors have in mind that $\mathcal{S}[X]=SXS^{-1}$ (I am not sure whether this can be true?), with $S$ an appropriate matrix, and that they are a bit confusing $S$ and $\mathcal{S}$ in the presentation. Or maybe they have always in mind the notation in the doubled space but they mix up the notation in section 2 where the doubled space has not being introduced.

If $\mathcal{S}$ is instead just a matrix as $\rho$ and not a super-operator like the Lindblad I cannot make sense of Eq.(2) as it is written now.

Another way to make everything clear could be to use, from the very beginning, a representation of the Lindblad operator as a matrix and of the density matrix as a vector. In this way, no ambiguity on the role of S can emerge. This is indeed what the authors do in the examples.

I also notice that, in section 3, Eq.(17) seems to say that the generator of the flow is actually a super-operator since it is given by the commutator of the two super-operators. As such, it looks like $\mathcal{S}$ should be a super-operator.

I believe that this "issue" with the notation could generate confusion. I would advise the authors to make this clear.

After this is done, in my opinion the paper certainly deserves to be published in Scipost physics.

I thank the authors for their detailed response and the changes made. I recognise that the authors have made some effort to improve the weaknesses pointed out in the previous report.
I thus recommend the manuscript for publication in SciPost Physics.

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