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Information Scrambling with Conservation Laws

by Jonah Kudler-Flam, Ramanjit Sohal, Laimei Nie

Submission summary

As Contributors: Jonah Kudler-Flam · Laimei Nie · Ramanjit Sohal
Arxiv Link: (pdf)
Date submitted: 2021-08-05 02:37
Submitted by: Sohal, Ramanjit
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as stochastic systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic perspective. For general non-integrable models, we use the equilibrium approximation to show that the maximal amount of information is scrambled (as measured by the tripartite mutual information of the time-evolution operator) at late times even when a system conserves energy. In contrast, we explicate how when a system has additional symmetries that lead to degeneracies in the spectrum, the amount of information scrambled must decrease. This general theory is exemplified in case studies of holographic conformal field theories (CFTs) and the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D CFTs, we argue that, in a sense, these holographic theories are not maximally chaotic, which is explicitly seen by the non-saturation of the second R\'enyi tripartite mutual information. The roles of particle-hole and U(1) symmetries in the SYK model are milder due to the degeneracies being only two-fold, which we confirm explicitly at both large- and small-$N$. We reinterpret the operator entanglement in terms the growth of local operators, connecting our results with the information scrambling described by out-of-time-ordered correlators, identifying the mechanism for suppressed scrambling from the Heisenberg perspective.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-9-25 (Invited Report)


1- Timely topic.
2- New perspective.


1- The equilibrium approximation seems too strong for locally interacting systems.


The paper studies the tripartite operator mutual information (TOMI), a well-established measure of scrambling of quantum information, in systems with degeneracies in the spectrum. The starting point is an observation made in Ref. [15]: surprisingly, holographic CFTs do not saturate the well-known lower bound for the TOMI, indicating that they do not scramble quantum information maximally. In the paper the authors use the so called "equilibrium approximation" to argue that this is due to the large degeneracies in the spectrum of CFTs (also holographic ones) imposed by the structure of the Virasoro algebra. They propose degeneracies in the spectrum as a general mechanism to limit the scrambling of quantum information. To substantiate this picture they show that in the SYK model (and its charged variant), where the degeneracy is only twofold, the bound is saturated in at large $N$ (number of particles). Finally, they use their results to give a different perspective on the operator growth.

I think that the paper is interesting, it presents an overall clear and comprehensive discussion, and represents a non-trivial addition to the existing literature. Therefore, I am inclined to recommend publication in Scipost. Before that, however, the authors should clarify better the main approximation underlying their work, i.e. the equilibrium approximation. With that approximation they require the field configurations on one spacetime sheet to exactly coincide with that on another (backward evolving) spacetime sheet for all $x$ and $t$. Isn't this too strong? I would expect that also configurations where this happens only locally in the space-time would contribute non-trivially. Namely I would expect something like $\phi_i(x,t)=\phi_{\sigma_{x,t}(i)}(x,t)$ where also the permutation depends on the (coarse grained) space-time point. This is, for example, what happens in the discrete analogue of the problem discussed in Ref. [31]. Of course this complicates a lot the analysis: The problem becomes that of calculating the partition function of a Stat. Mech. model in 2d, rather than than in 0d. I can understand that this approximation can work for the SYK (as the authors show), but I struggle to believe that it is good enough for locally interacting systems.

Requested changes

The authors should address the point above and the following list of minor points.

1- In the introduction the authors refer to quantum circuit models as ``stochastic". Even if it's true that most studies have been conducted in random quantum circuits, which are indeed stochastic, this is not necessarily the case (for example the "dual-unitary" circuits studied e.g. in Ref.~[28] are not necessarily stochastic). I would suggest to use the term: periodically driven. Note that this is enough not to have energy conservation, which is the property the authors are interested in.

2- Why the regions $C$ and $D$ have to be semi-infinite in the discussion after Eq. 31?

3- I don't understand the claim "The only dynamical input thus far is that we are working with theories that have sufficiently complex energy spectra in order to decohere at late times. This situation is generic and should only be violated by integrable systems." made at the beginning of Sec. 3. Integrable systems (even free systems!) generically have a complex enough spectrum to decohere, see e.g.

Essler and Fagotti, J. Stat. Mech. (2016) 064002.

The only difference is that the diagonal ensemble (or thermomixed double) will have different properties, i.e. will be equivalent to a generalised Gibbs ensemble.

4- I am not sure the authors explicitly say that $\ell_A=\ell/2$ in Eq. 67 and in the inline equation before it.

5- Few typos:
"of" missing in the last sentence of the abstract.

"." should be "," in Eq. 41

"subscripts" $\mapsto$ "superscripts" in the sentence before Eq. 134.

  • validity: good
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2021-9-12 (Invited Report)


1- detailed discussion of scrambling and several new results in physically relevant situations.
2- investigation of the effect of conserved quantities on scrambling.


The authors investigate the scrambling of information in several physically relevant situations, namely holographic conformal field theories and the Sachdev-Ye-Kitaev model. The authors focus on the situation in which the system possesses a conservation law. They argue that while energy conservation is compatible with maximal scrambling of the information, the presence of a U(1) conserved quantity lowers the amount of scrambled information. They also discuss the scrambling of operators.

Understanding the scrambling of information is an interesting and challenging topic that has attracted the attention of several communities. The authors address this problem by using several of the techniques that are currently available. Their results are scientifically sound and interesting. I think that this paper represents a useful addition to the present literature and I recommend its publication in Scipost Physics

I have some comments regarding the relation between this work and the literature:

1) Scrambling of information is known to appear also in integrable systems. This is because although in integrable systems the locality of information is preserved by the presence of well-defined quasiparticles, these have a nonlinear dispersion, unlike CFT systems. The authors could try to link their work to the results presented in

Phys. Rev. B 100, 115150 (2019)

2) The authors use the so-called equilibrium approximation to obtain the Renyi entropies of a state and of the thermofiele double. This amounts to saying that the Renyi entropies are the same as the Renyi entropies of the equilibrium state (formula (18) in the paper). This result holds generically also for integrable systems where there is an extensive number of conservation laws. The authors could mention

Phys. Rev. B 96, 115421 (2017)

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: excellent

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