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Random Matrix Theory of the Isospectral twirling
by Salvatore F. E. Oliviero, Lorenzo Leone, Francesco Caravelli, Alioscia Hamma
Submission summary
As Contributors:  Salvatore Francesco Emanuele Oliviero 
Arxiv Link:  https://arxiv.org/abs/2012.07681v2 (pdf) 
Date submitted:  20201221 11:09 
Submitted by:  Oliviero, Salvatore Francesco Emanuele 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this paper, we present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis of quantum manybody systems: Frame potentials, Loschmidt Echos, OTOCs, Entanglement, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CPmaps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones.
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Reports on this Submission
Report 1 by Karol Życzkowski on 2021111 Invited Report
Strengths
several new results on outoftimeorder correlators, LoschmidtEchos
and several measures of entanglement
for various assumptions concerning the ensembles of random matrices applied
Weaknesses
Long paper with so many results that it is difficult to appreciate which of them are the most important...
Report
The paper by Oliviero et al presents an analysis if the twirling operation from the point of view of the theory of random matrices. Isospectral twirling is often used in quantum information processing
as it describes the Haar average over multiple usage of a channel.
The paper presents a detailed analysis of
outoftimeorder correlators, LoschmidtEchos
and several measures of entanglement
for various assumptions concerning the
ensembles of random matrices applied.
The paper is well written and its results are presented clearly.
I do appreciate the Tables 1 and 2 which summarize well results obtained.
As they seem to be correct and new, so my opinion this work can
become suitable for the publication in SciPost.
In general the paper contains a lot of new and interesting results,
but their number seems to be even too large: thus the paper is really
long, difficult to read and appreciate.
To be honest I was not able to check
all the formulae and derivations presented...
Requested changes
My detailed remarks to the paper include:
a) p.1 The authors write:
>quantum chaos has traditionally been associated
> to salient properties of those systems which were quantized from classical > chaotic systems. Such systems were found to possess statistics of energy > levels spacings corresponding to the predictions
> of Random Matrix Theory (RMT) [3–14].
This is correct, but we know also a lot concerning the statistical properties of eigenvectors of chaotic systems, what could be mentioned here with appropriate references.
Another welldeveloped path of research concerned
quantum Lyapunov exponent and quantum dynamical entropies.
b) p. 4, U= exp  iHt  brackets are missing
c) p 9. Figure 1 (with known figure) is somehow misleading
 would you consider adding both axes, which cross the origin (0,0)
d) this figure (and all othee figures !)
 larger Figure labels and axes captions are welcome!
So small fonts of the labels make the data presented
difficult to understand
(and some labels are hardly legible
thus the figure becomes not readable)
e) p. 9 The authors discuss nearest level distributions for
Poisson, GOE and GUE and recall eq. (13).
the sentence
>The nearest level distributions for these distributions are given by...
sugests that Eq (13) presents exact results for large systems size which is not true  these are only results obtained for d=2
which can serve as reasonably precise approximations for the
asymptotic distributions, d> infty, see the book of Mehta [8].
This remark concerns also eq (123), p. 37.
f) p. 10) >>With the use of Levy lemma...
a good reference to the leve lemma would be helpful
g) p.12 Fig.2 presents the spectral functions c_2 and c_4
for Poisson ensemble, GDE and GUE. After looking at Fig 1
a reader might ask, how it looks like for GOE?
Are the differencies between GOE and GUE relevant here
(and in further quantities discussed later in the paper?
The same question concerns also Figs. 311).
h) p.13  the formulation
>>A tdesign is therefore ‘random enough’ up to the tth moment.
seems to me misleading, as the actual design is 'not random'  the points are carefully chosen to be placed 'uniformly' all around the space of all unitary matrices, such that the moments indeed agree with the integral
over the Haar measure...
Take the GaussLegendre approximate integration rule in interval [1,1]
approximated by the value of the function in two points $x_{\pm}=\pm 1/sqrt{3}$  see
https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature
Would you call these numbers 'random'?
The reasoning goes in other direction as
a set of random numbers (or quasirandom  produced by a chaotic dynamics!)
can approximate a $t$design up to a certain accuracy.
i) p. 20, after eq (44) we learn that
>Eq. (44) converges to the equilibrium value in a time
> O(d^{1/12}) and O(d^{1/8}) for GUE and Poisson, respectively.
is it simple to explain the reader where these two exponents came from?
(this remark concerns also the caption to Fig 6 and 9.)
j) I am not sure I understand Eq.(59) correctly 
perhaps the last exponent should go after the last bracket
to have
tr (D_B(\sqrt{rho}))^2
as square is taken after the dephasing operator D_B...
is it so?
k) what is the physical meaning of the operator {\cal K} in eq (84).
Is it somehow related to the superoperator (e.g. by partial transpose?)
So it is also linked by another change of entries
of the matrix to the Choi matrix D of the operation Q?
Just after Eq. (84) a statement appears:
>> taking T as the swap operator, then tr (TK) = d.
is this just a consequence of the trance preserving condition
which implies the standard normalization of the Choi matrix,
Tr D=d ?
If this is so, such a remakr could be helpful for a reader
to see where this property comes from...
l) The sentence in p. 62 above Eq. (275) reads
The action of a unital map can be written as
f(rho) = \sum_k U_k rho U_k^ (275)
Any map in the above form of a mixed unitary channel is unital
but the reverse statement is true for d=2 only.
For d=3 there exist the channel of LandauStreater
which is unital but is not mixed unitary as (275).
Thus it looks like this proof works for d=2 only
and for higher dimensions in requires improvements...
m) well known papers of Wigner and Dyson
have several parts, which are labeled by Roman numbers
say paper I and paper II (capital letters  ref [5] and [7])
n) the authors might consider quoting in Ref. [9] the more recent edition
of the book by Haake  revised and considerably extended
o) Ref [22] has title
Hidden correlations in the hawking radiation
possibly capital letter: Hawking,
also Neumann's in [76]
and Loschmidt in ref [86]