The complexity of identifying Ryu-Takayanagi surfaces in AdS3/CFT2

Bao, Ning; Chatwin-Davies, Aidan

doi:10.1007/jhep11(2016)034

Cited by 3 publications

(10 citation statements)

References 24 publications

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“…While this is well-known and straightforward for the n = 2 case just discussed, some interest has recently emerged [54,[80][81][82][83] for similar concepts for situations involving n > 2 disconnected intervals. Here we present our study of this case, and apply it to the steady-state spacetime in subsection 6.4.…”

Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 97%

“…The Wolfram Mathematica code that we use for this analysis is uploaded to the arXiv as an ancillary file together with this paper and with a sample of the numerical results that is obtained from the matching procedure of section 4. 20 There is some overlap between the issues addressed in this section (especially subsection 6.1) and the ones investigated in [83], which was published after most of this section was completed. Although we are working in a covariant (time-dependent) setting, the findings of [83] suggest that the code used in our ancillary file may still be optimized.…”

Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 99%

“…20 There is some overlap between the issues addressed in this section (especially subsection 6.1) and the ones investigated in [83], which was published after most of this section was completed. Although we are working in a covariant (time-dependent) setting, the findings of [83] suggest that the code used in our ancillary file may still be optimized. However, it nevertheless produces the desired results.…”

Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 99%

“…However, in this subsection we provide an algorithm that for any n enumerates the possible phases in a consistent manner, without omitting any solution or counting it twice, and which can easily be implemented (see the corresponding ancillary file). We do not assume a translation invariant spacetime, however we will assume a spacetime with simple topology, such as Poincaré AdS or a flat black brane, excluding possible phenomena such as entanglement plateaux [85], see also [83]. Our task then essentially boils down to finding the noncrossing partitions of a set with n elements, a well known combinatorial problem related to the Catalan numbers C n [86].…”

Section: The Phases Of the Union Of N Intervalsmentioning

confidence: 99%

“…which grows as C n ∼ 4 n n 3/2 √ π for large n. However, with a more optimized code, it may not be necessary to compute all the values of this number of phases [83]. 21 Remember that the numbers 1 to 6 serve here as labels of (ordered) boundary points, and not necessarily as coordinates on the x-axis.…”

Section: The Phases Of the Union Of N Intervalsmentioning

confidence: 99%

See 4 more Smart Citations

Time evolution of entanglement for holographic steady state formation

Erdmenger

Fernández

Flory

et al. 2017

J. High Energ. Phys.

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Within gauge/gravity duality, we consider the local quench-like time evolution obtained by joining two 1+1-dimensional heat baths at different temperatures at time t = 0. A steady state forms and expands in space. For the 2+1-dimensional gravity dual, we find that the "shockwaves" expanding the steady-state region are of spacelike nature in the bulk despite being null at the boundary. However, they do not transport information. Moreover, by adapting the time-dependent Hubeny-Rangamani-Takayanagi prescription, we holographically calculate the entanglement entropy and also the mutual information for different entangling regions. For general temperatures, we find that the entanglement entropy increase rate satisfies the same bound as in the 'entanglement tsunami' setups. For small temperatures of the two baths, we derive an analytical formula for the time dependence of the entanglement entropy. This replaces the entanglement tsunami-like behaviour seen for high temperatures. Finally, we check that strong subadditivity holds in this time-dependent system, as well as further more general entanglement inequalities for five or more regions recently derived for the static case.

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Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 97%

Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 99%

Section: Entanglement Entropies Of Systems With Many Disconnected Commentioning

confidence: 99%

Section: The Phases Of the Union Of N Intervalsmentioning

confidence: 99%

Section: The Phases Of the Union Of N Intervalsmentioning

confidence: 99%

See 3 more Smart Citations

Time evolution of entanglement for holographic steady state formation

Erdmenger

Fernández

Flory

et al. 2017

J. High Energ. Phys.

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Discrete bulk reconstruction

Aaronson

Pollack

2023

J. High Energ. Phys.

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According to the AdS/CFT correspondence, the geometries of certain spacetimes are fully determined by quantum states that live on their boundaries — indeed, by the von Neumann entropies of portions of those boundary states. This work investigates to what extent the geometries can be reconstructed from the entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued that the AdS/CFT map can be exponentially complex if one wants to reconstruct regions such as the interiors of black holes. Our main result provides a sort of converse: we show that, in the special case of a single 1D boundary divided into N “atomic regions”, if the input data consists of a list of entropies of contiguous boundary regions, and if the entropies satisfy a single inequality called Strong Subadditivity, then we can construct a graph model for the bulk in linear time. Moreover, the bulk graph is planar, it has O(N2) vertices (the information-theoretic minimum), and it’s “universal”, with only the edge weights depending on the specific entropies in question. From a combinatorial perspective, our problem boils down to an “inverse” of the famous min-cut problem: rather than being given a graph and asked to find a min-cut, here we’re given the values of min-cuts separating various sets of vertices, and need to find a weighted undirected graph consistent with those values. Our solution to this problem relies on the notion of a “bulkless” graph, which might be of independent interest for AdS/CFT. We also make initial progress on the case of multiple 1D boundaries — where the boundaries could be connected via wormholes — including an upper bound of O(N4) vertices whenever an embeddable bulk graph exists (thus putting the problem into the complexity class NP).

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Complexity growth in massive gravity theories, the effects of chirality, and more

Ghodrati

2017

Phys. Rev. D

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To study the effect of parity-violation on the rate of complexity growth, by using "Com-plexity=Action" conjecture, we find the complexity growth rates in different solutions of the chiral theory of Topologically Massive Gravity (TMG) and parity-preserving theory of New Massive Gravity (NMG). Using the results, one can see that decreasing the parameter µ, which increases the effect of Chern-Simons term and increases chirality, would increase the rate of growth of complexity. Also one can observe a stronger correlation between complexity growth and temperature rather than complexity growth and entropy. At the end we comment on the possible meaning of the deforming term of chiral Liouville action for the rate of complexity growth of warped CFTs in the Tensor Network Renormalization picture.

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The complexity of identifying Ryu-Takayanagi surfaces in AdS3/CFT2

Abstract: We present a constructive algorithm for the determination of Ryu-Takayanagi surfaces in AdS 3 /CFT 2 which exploits previously noted connections between holographic entanglement entropy and max-flow/min-cut. We then characterize its complexity as a polynomial time algorithm.

Cited by 3 publications

References 24 publications

Time evolution of entanglement for holographic steady state formation

Time evolution of entanglement for holographic steady state formation

Discrete bulk reconstruction

Complexity growth in massive gravity theories, the effects of chirality, and more

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