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Equilibration of quantum cat states
by Tony Jin
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Submission summary
As Contributors:  Tony Jin 
Arxiv Link:  https://arxiv.org/abs/2003.04702v2 (pdf) 
Date submitted:  20200410 02:00 
Submitted by:  Jin, Tony 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states. Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state. We backup our analysis with numerical results obtained on the XXX spin chain with a random field along the zaxis in the ergodic regime and find good qualitative and quantitative agreement with the theory. We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.
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Anonymous Report 2 on 202061 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.04702v2, delivered 20200601, doi: 10.21468/SciPost.Report.1727
Strengths
1  Rather fresh and original approach to a very timely problem
2  The main statements and definitions concisely formulated
3  Presentation has a very good balance between heuristic presentation and formal derivations and proofs (mainly in appendices)
Weaknesses
1  Presentation of numerics is a bit "too sketchy" and imprecise
2  Text needs a bit more of (language) polishing
Report
This is a very nice paper and I strongly recommend its publication. It presents an approach to quantum ergodicity from the viewpoint of fluctuations of physical observables. After a careful thought, the result may not be very surprising, but it is very nicely and clearly formulated. First of all, even the standard definition of quantum ergodicity is given in very general and flexible formulation, allowing for any number of conserved quantities.
The author then generalizes the ergodictimeaveaging projector to fluctuating observables which is formulated in a tensor product Hilbert space
of ncopies of the original system. Not only the result for the second moment (n=2) is compactly written down, also the general result for any moment order n is derived in appendix. What is remarkable is, that this timeaveraging projectors in higher tensorproduct spaces are never really ergodic (even for a nicely ergodic system) but they generically keep memory of the initial state. This is demonstrated in a numerical example of XXX chain in a random field, starting from initial cat states.
I particularly find interesting the discussion at the end in relation to integrable systems. Specifically, the presentation as it is formulated refers to systems with finite Hilbert spaces, where one can formulate ergodicity through spectral decomposition. It would be interesting to think of generalizing this to truly extended (infinite) quantum systems, where locality or nonlocality of conserved operators would become important.
Requested changes
1  Discussion of numerical example is a bit imprecise (or sketchy). For example the author refers to "mean eigenvalue spacing" as the `mean ratio of consecutive level spacings'. Please be more precise, as the mean level spacing is an irrelevant quantity.
2  It is not clear in Figure 2 what is meant by error bars (mentioned in the main text)? I guess there is no "disorder averaging" in numerics, but what is the meaning of the blueshaded region in the top right panel?
3  The meaning of operatorabsolute value (Q) in the last display equation at the end of page 6 is not explained.
4  In discussion of ETH is section 4 it is not clear why ETH  which is a statement about the distribution of matrix elements of local observables  should depend on the initial state (of course it does through \delta E, in particular when one would want to apply it to cat states where no longer \delta E << E, but I guess this should be explicitly stated).
Anonymous Report 1 on 2020516 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.04702v2, delivered 20200516, doi: 10.21468/SciPost.Report.1690
Report
Jin's work addresses an interesting question, how can one identify cat states after a system has equilibrated? He shows that the time fluctuations in the steady state distinguish cat states from other more traditional states.
The results reported in the paper look technically correct but I think the presentation can be improved. I found some statements that may not be correct. Below I point out some of my concerns and questions in the order they appear in the manuscript. They are not ranked by importance.
1) On page 2, I do not understand this sentence: "We will see that such states present non trivial, possibly nonlocal fluctuations of the offdiagonal components in the steadystate that are fixed by the initial quantum coherences." The "offdiagonal components" of what?
2) On page 4, what is the "diagonal ensemble"? I also do not understand why the case in which the Hamiltonian is the identity operator ("one energy sector that is the whole Hilbert space") is called the "usual microcanonical ensemble." If the Hamiltonian is the identity operator there is no dynamics so I do not see the point of highlighting that "extreme case".
3) Throughout the manuscript instead of the word "moment" the author incorrectly uses the word "momenta". The first place where I identified this was in the first line on page 5.
4) I suggest the author to number all the equations. Below I have a comment about an equation that is not numbered.
5) I would not call the state for protocol II (unnumbered equation) a "classically mixed state", and I would not call $\hat Q$ an observable. $\hat Q$ is a highly many body operator. Has the author found similar signatures to the ones in the right panels in figure 2 in a local operator?
6) On page 8, what the author describes as the ETH is not what I understand as the ETH. I understand the ETH as a statement about matrix elements of observables that has nothing to do with the initial states. In that sense the discussion in the last paragraph of page 8 does not appear to be an equivalent statement to the ETH but just the ETH applied to the states the author is studying. I should add that $\hat Q$ does not look to me like an operator that would fulfill the ETH.
7) On page 9, the author writes "Typicality states that for all eigenstates of the energy window, fewbody operators have thermal distributions in the thermodynamic limit." This is not what I understand as typicality, which I don't think says anything about eigenstates but rather about pure states that are random superpositions of eigenstates in the energy window.
8) Also on page 9 the author writes: "Typically, for finitesize integrable systems for example, one expects the existence of longlived oscillations that prevents the system from equilibrating [30, 31]." I don't think this is what generally happens in finitesize integrable systems, which I believe equilibrate so long as they are not too small or the initial state is not too special. There is a long literature on this involving the generalized Gibbs ensemble in finitesize integrable systems.
Author: Tony Jin on 20200529 [id 842]
(in reply to Report 1 on 20200516)I thank the referee for very useful comments and criticism. I have not yet made changes appear in the preprint as I am waiting for the editor’s recommendation to do so. Let us address the different points one by one :
1) On page 2, I do not understand this sentence: "We will see that such states present non trivial, possibly nonlocal fluctuations of the offdiagonal components in the steadystate that are fixed by the initial quantum coherences." The "offdiagonal components" of what?
The sentence may be not precise enough indeed : Here it is implicit that everything refers to the eigenbasis of the Hamiltonian. The offdiagonal components are offdiagonal elements of H. I will add a sentence to clarify this point.
2) On page 4, what is the "diagonal ensemble"? I also do not understand why the case in which the Hamiltonian is the identity operator ("one energy sector that is the whole Hilbert space") is called the "usual microcanonical ensemble." If the Hamiltonian is the identity operator there is no dynamics so I do not see the point of highlighting that "extreme case".
The diagonal ensemble is the ensemble where only the diagonal elements of the density matrices have been kept and the offdiagonal have been set to zero. I will add a sentence to make this definition precise. What I call the usual microcanonical ensemble is the normalized identity matrix written in the Hamiltonian eigenbasis. I highlighted these two cases because it’s an important question in thermalization to know how to go from a system described by the diagonal ensemble to the microcanonical ensemble. What is expected from averaging over timeevolution of ergodic quantum systems is that density matrices are given by the diagonal ensemble. To get that the microcanonical ensemble produces the correct output for the expectation values of observables, one needs additional hypothesis for instance on the distribution of the elements of observables of interest. (See for instance the reference [14] for a discussion of this in the context of random matrix theory and ETH). I discuss how this correspondence can be done in the framework of the paper in the section 4. I will add a sentence to make this motivation clear.
3) Throughout the manuscript instead of the word "moment" the author incorrectly uses the word "momenta". The first place where I identified this was in the first line on page 5.
Thank you for pointing this out. It will be corrected.
4) I suggest the author to number all the equations. Below I have a comment about an equation that is not numbered.
Ok.
5) I would not call the state for protocol II (unnumbered equation) a "classically mixed state", and I would not call Q an observable. Q is a highly many body operator. Has the author found similar signatures to the ones in the right panels in figure 2 in a local operator?
I guess it’s only a semantic problem but here classically mixed state means that protocol II is just a classical statistical superposition of \Psi_1 and \Psi_2 with no quantum coherence. Maybe “mixed state” would be a better terminology. What I call an observable is just a selfadjoint operator in the Hilbert space. Q is indeed a highly non local operator and in general no suited for experimental applications. I will add a note to make this precise. In principle, similar statements concerning the fluctuations hold for local operators as long as they have non zero projection on the off diagonal sector (last line of eq(3)). For instance one could imagine to make an interference experiment between two distant parts of the system. In average the interference should be zero for both protocols but the amplitude of the fluctuations should be greater for protocol I. I will add a few sentences to discuss this local versus non local observables.
6) On page 8, what the author describes as the ETH is not what I understand as the ETH. I understand the ETH as a statement about matrix elements of observables that has nothing to do with the initial states. In that sense the discussion in the last paragraph of page 8 does not appear to be an equivalent statement to the ETH but just the ETH applied to the states the author is studying. I should add that Q does not look to me like an operator that would fulfill the ETH.
I am not sure to see where we disagree, I also state that the ETH is a statement about matrix elements (first paragraph of p 8). I do think though that one has to fix a thin energy shell from which one draws eigenstate upon which expectation values are computed (the energy scale E has to come from somewhere for instance). What I am saying in this part is that we can replace the assumption made in the ETH on the distribution of matrix elements of observables by an assumption on the group upon which the dynamics is invariant. Both assumptions yield the same expectations for observables. I also agree on the fact that Q, by its nonlocal nature should not fulfill the ETH, hence the need for a more general theory.
7) On page 9, the author writes "Typicality states that for all eigenstates of the energy window, fewbody operators have thermal distributions in the thermodynamic limit." This is not what I understand as typicality, which I don't think says anything about eigenstates but rather about pure states that are random superpositions of eigenstates in the energy window.
This is indeed a more accurate statement. It will be corrected.
8) Also on page 9 the author writes: "Typically, for finitesize integrable systems for example, one expects the existence of longlived oscillations that prevents the system from equilibrating [30, 31]." I don't think this is what generally happens in finitesize integrable systems, which I believe equilibrate so long as they are not too small or the initial state is not too special. There is a long literature on this involving the generalized Gibbs ensemble in finitesize integrable systems.
I think you are correct. I will modify this statement to say that longlived oscillations can happen even though this may not constitute the most general situation. Also I believe relaxation towards GGE holds for local observables but here I address the equilibrium state of the whole system.