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Local Operator Entanglement in Spin Chains
by Eric Mascot, Masahiro Nozaki, Masaki Tezuka
Submission summary
As Contributors:  Eric Mascot 
Arxiv Link:  https://arxiv.org/abs/2012.14609v3 (pdf) 
Date submitted:  20210120 08:02 
Submitted by:  Mascot, Eric 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the time evolution of bi and tripartite operator mutual information of the timeevolution operator and Pauli's spin operators in the onedimensional Ising model with magnetic field and the disordered Heisenberg model. In the Ising model, the earlytime evolution qualitatively follows an effective light cone picture, and the latetime value is well described by Page's value for a random pure state. In the Heisenberg model with strong disorder, we find manybody localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian describes the time evolution of bi and tripartite operator mutual information for the Heisenberg model in the large disorder regime.
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This paper presents numerical data for entanglement measures in a chaotic and a disordered chain, namely the mutual information and the “tripartite mutual information” for timeevolving operators. It is clear that considerable work has gone into the generation of the data and the presentation of this data and of background material. However, this manuscript does not appear to me to be publishable in Scipost. What is lacking is (a) a clearly defined motivation for studying these quantities and (b) clear lessons from the numerical data. The reader is shown many plots without necessarily knowing what to infer from them.
The authors do make attempts to interpret some of the findings (they describe a “light cone picture”, they use some basic properties of strongly entangled states, and they invoke the lbit picture for manybody localization in the disordered case). However these theoretical accounts seem relatively limited. For example the discussion of the chaotic phase is not quantitative (except in the relatively simple t>∞ limit).
The observables considered are linear combinations of operator entanglement entropies. Therefore in the chaotic phase existing ideas can be used to give a more quantitative account of their behavior, to the extent that this is universal at large scales. The scaling theory for operator entanglement in chaotic chains of Ref 19 applies here.
Regardless of the presence or absence of a theoretical explanation, clearer physical motivation for studying these quantities is needed. If I understand correctly, the idea behind the tripartite information of a channel in previous work was to determine whether information that is initially encoded locally is encoded locally or globally after the application of the channel. In the present paper the context is different (local operators) and the motivation is less clear to me. It is true that the local operators studied here happen to be unitary, so they can be viewed as channels if desired, but the authors do not give an argument that it is useful to do this.
The paper is currently rather long. The authors might wish to decide what the key message(s) of their paper should be, and cut down the length and the number of figures in the main text to give the paper some focus.
Here are more specific comments.
Eq 1  I understand the general idea the authors are trying to convey, but this equation is not correct and neither is the claimed relation between energy and temperature, in general.
Eq 3  missing normalization
Eq 5  can this be reformulated to be clearer?
Eq 9  this quantity is claimed to be “exponentially large”  isn’t it bounded and of order 1 in the models studied here?
Sec 2.2. “we expect quantum information to be preserved locally”. Generically only classical information is preserved locally in the MBL phase (the zcomponent of the lbit is conserved but not the xcomponent).
p6 “robust against noise” What is meant here? Noise (timedependent fluctuations in the Hamiltonian) will generically destroy the MBL phase.
“linear in noninteracting systems or general nonintegrable systems without localization”. This statement assumes translational invariance. Otherwise, sublinear growth is possible, even without localization.
p9 “unless O is the identity operator” This is not correct, there are other O with this property
p10, 11 contain many vague (e.g. “unrelated to each other”, “data regarding the local entanglement structure”) or repetitive statements
p19 should S_A,B be I_A,B?