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Page Curve for Entanglement Negativity through Geometric Evaporation

by Jaydeep Kumar Basak, Debarshi Basu, Vinay Malvimat, Himanshu Parihar and Gautam Sengupta

Submission summary

As Contributors: Jaydeep Kumar Basak · Debarshi Basu · Vinay Malvimat · Himanshu Parihar
Preprint link: scipost_202107_00042v1
Date submitted: 2021-07-21 17:07
Submitted by: Malvimat, Vinay
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We compute the entanglement negativity for various pure and mixed state configurations in a bath coupled to an evaporating two dimensional non-extremal Jackiw-Teitelboim (JT) black hole obtained through the partial dimensional reduction of a three dimensional BTZ black hole. Our results exactly reproduce the analogues of the Page curve for the entanglement negativity which were recently determined through diagrammatic technique developed in the context of random matrix theory.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-9-4 (Invited Report)


The connection between black hole and random matrix theory.


- The relation between the area of back-reacting cosmic branes in AdS$_3$ and geodesic lengths is valid only for spherical entangling surfaces.

- The manuscript lacks of clarity in some part.


In this work, the authors compute holographic entanglement negativity and analyse its dependence in terms of the relevant scales in the system (referred to as analogues of Page curve for negativity) for pure and mixed states in a bath coupled to a two-dimensional Jackiw-Teitelboim (JT) black hole.

To be more specific, the system is holographically engineered via a ``dimensional reduction'' proposed by Verheijden et al. (ref. [20] in the manuscript), where a JT black hole is obtained after a partial dimensional reduction of a three-dimensional BTZ black hole. The rest of the three-dimensional geometry is dual to a conformal field theory (CFT$_2$).
The crucial points in ref [20] for this manuscript are a) the dynamical evaporation of a JT black hole is realised by making the parameter controlling the reduction time-dependent, and b) dynamical quantities like entanglement of radiation can be simply computed from a three-dimensional geometrical point of view, as shown already in ref. [20].

Entanglement negativity (or logarithmic negativity) is an entanglement measure suitable for mixed states, it has the benefit to detect only quantum correlations and to be computable in quantum field theory settings.
Here, holographic entanglement negativity (HEN) is computed according to a series of conjectures originally proposed in ref. [40, 43, 45] in the manuscript.
In essence, the authors expressed the HEN in terms of the dual of Renyi entropies of order $1/2$, which are given by the minimum of geodesic lengths computed according to the dictionary proposed in ref. [20].

Equipped with this, the authors investigate the analogues of a Page curve for mixed states.
Their results qualitatively match the findings of Shapourian et al. (ref. [54] in the manuscript) for random mixed states.
There are various scales to be tuned here (e.g. the relative size between the subsystems $A_i$, the relative size between the whole subsystem $A$ and the bath $B$, the distance between the subsystems $A_i$ in the disjoint case). The resulting phase diagram of negativity as a function of these scales is rather reach and can distinguish between different phases, depending on the parameters which are kept fixed or not. The authors describe the different cases in detail.

The problem faced in the manuscript and the methods used to compute entanglement negativity are not new, however the authors are able to add a novel and interesting result to this very active line of research by combining previous techniques.
Holographic entanglement negativity in the bath with a coupled JT black holes was already discussed in a previous paper by Kumar Basak et al. (ref [48] in the manuscript). Here the novelty is the use of the setting proposed in ref.[20] and the systematic investigation of an analogue of a Page curve for the negativity.

In my opinion the work contains some interesting material which deserves to be published.
Before publication, the authors should address the following points.

Requested changes

1. Introduction. Page 1 and largely page 2 are devoted to review previous results. However these pages contain information which are not directly used in the actual analysis. In my opinion, the reader would benefit from a more streamed and concise introduction. Moreover, the novel aspects on the manuscript are summarised in 5 lines on page 4. This section is "unbalanced' and misguide the reader. I understand this is a general comment, but I strongly advice to improve the narrative of the introduction, for example I do not see the reason to describe so extensively results of ref. [48], when their techniques and results are not directly used in this work.

2. Section 2.3, page 8. Below eq. (19) the authors are referring to the wrong paper, I believe ref [45] (and not [43]) should be mentioned.

3. Section 2.3. The authors list eq. (19) and (21) which are the expressions used later to computed the negativity. For reader's convenience it would be useful to report here also the analogue expression for adjacent intervals.

4. Page 10, eq. (28) shows the minimum of four configurations and not three as said above eq. (28).

5. Overall, on the plots not all the numerical values are indicated, for example $L$ is missing.

6. All the plots are (consistently) obtained for the same values of certain parameters, such as $\Phi_0$ and $\beta$. Did the authors try other values of temperature and cutoff? If so, is there any qualitative difference?

7. I would like to have some clarifications to fig. 6. This seems essentially a four-partite system, four lengths are in play (though not all independent) and it is not directly analogue to the configurations examined in the random matrix theory as in ref. [53]. Could the authors comment on this?

8. Figures 11 and 6 are very similar, even though obtained for disjoint and adjacent intervals respectively. Usually in field theories logarithmic negativity for adjacent and disjoint intervals has different behaviours. Can the authors comment on this result?

9. Sections 3.1.4, and 3.1.5. Also, it seems that there is no qualitatively different behaviour when the JT black hole is involved, for adjacent and disjoint intervals. But how solid is this result? The authors are using the same numerical value for $b$ as in previous sections. Can the authors comment on this?

10. HEN is obtained as a certain linear combination of geodesic lengths, which, for spherical entangling surfaces, are proportional to back-reacting cosmic branes in AdS$_3$, which in turn are conjectured to be dual to Renyi entropies of order $1/2$.

The derivation of the expressions for the holographic entanglement negativity is somewhat confusing. It starts with the expression in terms of geodesic lengths, according to the proposal ref [40]. On the other hand the geodesic lengths are proportional to the area of back-reacting cosmic branes in AdS$_3$ (at least for certain symmetric configurations of the intervals), and these are related to Renyi entropies of order 1/2. Again these are expressed in terms of lengths computed according to the prescriptions of ref. [20]. Why do the authors cannot formulate directly the initial expression for example in eq. (14) in terms of eq. (3, 4) (assuming that we are taking the min)? It would be useful to see a clarification on this point.

11. Finally, there are various typos that should be corrected (punctuations and spaces, page $\to$ Page, algbraic $\to$ algebraic etc. etc.).

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2021-8-22 (Invited Report)


1- Some notations and concepts are not explained properly.
2-Do not provide enough analytical results.
3-Too many similar numerical plots but the conclusions and lessons from these plots are not clear.
4-The role of the dynamics of the evaporating BH in the entanglement negativity is not explained.


This manuscript studied the entanglement negativity of various subsystems in 2D JT gravity coupled to a bath. Particularly, the 2D model is proposed in Ref[20] and derived from the 3D BTZ black hole by performing a partial dimension reduction of the $ \phi$-direction. By making the reduction angle $\alpha$ time-dependent, it realizes the 2D black hole evaporation in a geometric way.

In this manuscript, the authors numerically studied the holographic entanglement negativity of adjacent intervals and also disjoint intervals. However, I think the authors should be able to provide more analytical results. So I would like to request the authors to add more calculations and resubmit the manuscript. My two main requests are
01- Add analytical results for the entanglement negativity of different systems. For a single interval, the Renyi entropy of order half has been shown in eq.(29).
The full results for $\mathcal{E}$ are expected to be complicated. However, most plots in this manuscript present very simple behaviors. Based on eqs.(29,30), the authors can explain the linear growth and derive the growth rate. And in some regimes, the authors should use their formula to illustrate the appearance of the plateaus.
02-In eq(33), the authors present the entanglement negativity with islands. Although the holography entanglement negativity is enough to derive the results, it is still worth exploring the island formula in detail. Of course, it is not easy to calculate the contributions $\mathcal{E}^{\mathrm{eff}}$. But the leading contribution is given by the area term. In this geometric evaporating BH model, the time evolution of the dilaton is solvable. This makes it possible to match the holographic result by using the island formula.

I have some other minor questions or suggestions as listed in "Requested Changes".

Requested changes

1- In section 2.1, the authors should add the expression for the time-dependent $\alpha(t)$ and explain why this can result in an evaporating black hole.
2- For the results shown in this manuscript, I do not understand what is the role of the evaporating BH. For example, most figures (eg, 6,9, 11,12,14,16) for the entanglement negativity correspond to a fixed value of $\alpha$. What is the influence of BH evaporation on the entanglement negativity of different subsystems?
3- The interval B should be labeled in the figures.
4- In equation (29), the authors introduce a parameter $\phi'$ as the distance between subsystem R and interval B. But I do not understand what is the meaning of this extra parameter. Is this used to regularize the limit $b \to 0$? But this is can be done by putting a cut-off as $b \to \phi'$. From eq(29), you can redefine $b \to b+\phi'$ to absorb that parameter $\phi'$.
5- For all the numerical plots with various parameters, I guess the authors fix the AdS radius as 1. I prefer to choose dimensionless parameters like $L/\beta, b/L$ in numerics.
6- In most captions of figures, the authors choose $c=500$. But this notation $c$ and its connection to $G_N$ is not introduced in the context and $c$ never appears in any formulas.
7- In figure 2, 3, the x-axis is $1-\alpha$. Do the authors want to use this as a time direction? From the evaporation model discussed in ref[20], it is related to the tilded time. See their eq(4.17)
\[\alpha(\tilde{t}) = 1- \frac{A}{2}\tilde{t}.\]
If this is the model the authors want to discuss, this point should be explained in the context.
8- Figure 5 is named as the Page curve for entanglement negativity. But I do not understand the meaning of this because it only represents a fixed time slice (a fixed $\alpha$).
9- According to the context after eq.(29), the subsystem R is parametrized by the angle $\phi_i$. But in the eq. (29), the angle $|\phi_2-\phi_1|$ represents a length like $b, L$. So the eq.(29) may have some typos or the explanation of $\phi_i$ is not correct. And I think it is not a good idea to use the angles and length $b, l_1, l_2$ in the meantime.
10-In eq.(33), the authors should explain what is the meaning of area $\mathcal{A}^{1/2}$ in 2D JT gravity and its connection to the value of dilaton because the surface $Q$ is just a point in this model.

  • validity: good
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: reasonable
  • grammar: good

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