确定AdS3/CFT2中Ryu-Takayanagi表面的算法与复杂性分析

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"这篇论文是关于在AdS 3 / CFT 2对偶理论中识别Ryu-Takayanagi表面及其复杂性的研究。作者提出了一个确定性的算法，该算法利用了全息纠缠熵与最大流量/最小切割原理之间的关系，以解决在反德西特时空（AdS 3）和其对应的共形场论（CFT 2）中的Ryu-Takayanagi公式问题。他们将这个算法的复杂性描述为多项式时间算法，这意味着它在计算效率上相对高效。 Ryu-Takayanagi公式是量子引力和弦理论中的一个关键概念，它提供了计算边界理论中特定区域纠缠熵的方法。这个公式将边界上的纠缠熵与AdS空间中的最小面积表面（即Ryu-Takayanagi表面）关联起来。通过找到这样的表面，可以定量地理解量子系统中的信息和纠缠。在论文的“算法来识别最小长度的bulk表面”部分，作者详细介绍了如何利用最大流量/最小切割的思想来找到这些表面。这是一种优化问题的经典方法，通常在图论和网络流问题中应用。在这个上下文中，它帮助在AdS空间中找到连接特定边界区域的最短路径。接下来，复杂性分析章节中，作者分析了他们的算法在处理不同规模问题时的效率。他们证明了算法的运行时间是多项式的，这意味着对于输入大小的增加，所需计算资源的增长速度并不快，这是计算复杂性理论中理想的情况。在“其他bulk拓扑”部分，作者可能探讨了在更复杂背景下的Ryu-Takayanagi表面的识别，如存在多区域纠缠或者非平凡几何结构的情况。这些情况可能会带来额外的挑战，但算法仍然能够有效地处理它们。最后的结论部分，作者总结了他们的发现，并可能讨论了这些结果对未来研究的影响，包括可能的应用于更广泛的AdS / CFT对偶问题以及对量子信息理论的理解。附录“A Geodesics的交点”可能提供了关于如何处理和计算AdS空间中测地线交点的额外细节，这在确定Ryu-Takayanagi表面时可能是必要的。这篇论文提供了一个实用且高效的工具，用于在AdS 3 / CFT 2框架下计算纠缠熵，深化了我们对量子引力和复杂性理论的理解。"

JHEP11(2016)034

union of n subregions, the brute force solution consists of checking every combination of n

geodesics that run between the 2n endpoints — a task that scales exponentially in n. Is

there a more eﬃcient way to identify the correct union of geodesics, or is the combinatorics

of boundary subregions a source of hardness even in a one-dimensional boundary?

Strikingly, we ﬁnd that there is a strong simpliﬁcation to a polynomial time algorithm

in three-dimensional gravity. Following previous inspiring and precise statements of a

connection between the Ryu-Takayanagi conjecture and max-ﬂow/min-cut [4, 5], we ﬁrst

devise a constructive algorithm that reduces the problem of determining the minimal-area

surface in the context of AdS

/CFT

to solving max-ﬂow/min-cut on a graph. Crucially,

the latter problem can be solved in polynomial time. We then analyze the computational

overhead that is required to reduce the problem to max-ﬂow/min-cut and verify that it

requires no more than polynomial time as well.

The organization of the paper is as follows: in section 2 we present the algorithm to

identify the bulk minimal surface in mathematical terms, and in section 3 we analyze the

complexity of this algorithm. In section 4, we discuss how our approach generalizes to

nontrivial bulk topologies. Finally, in section 5 we conclude with a few remarks.

2 An algorithm to identify minimal-length bulk surfaces

We begin by precisely stating the problem. Let X be a spatial slice of an asymptotically

AdS

spacetime, and suppose that X is simply-connected. Further, suppose that X is

holographically dual to a CFT state ρ deﬁned on its boundary, ∂X. Let {A

}

i=1

with n ≥ 2

be a collection of non-empty, simply-connected, closed, disjoint boundary regions, i.e., A

⊂

∂X, A

6= ∅, A

= cl(A

) for i ∈ [n],

and A

∩A

= ∅ for i 6= j. What is the Ryu-Takayanagi

surface, i.e., the minimal-length bulk surface

A that is homologous to A =

i=1

? Or,

if there are several minimal-length surfaces, what is one of them? Note that we may

consider strictly disjoint regions without loss of any generality; two overlapping or touching

regions A

and A

i+1

may be fused, since the area of any surface that subtends A

and A

i+1

separately can only be greater than or equal to the area of a surface that subtends A

∪A

i+1

Our approach to identifying

A is to reformulate the question as a problem on a graph

using a construction that is a variation on the one presented in section 3 of [5]. Without

loss of generality, suppose that the A

are numbered according to the order in which they

appear going counter-clockwise along ∂X and let A

= [a

, b

], again with respect to counter-

clockwise ordering. Next, for each a

, draw geodesics between it and every b

so that each

geodesic subtends a boundary region [a

, b

] whose interior contains zero or an even number

of boundary endpoints

(ﬁgure 1a). Because X is two-dimensional and simply-connected,

these geodesics are precisely the curves that could possibly make up the bulk minimal sur-

face, or in other words,

A is a subset of these geodesics. Since each a

is connected to each of

the n endpoints b

, there are n

geodesics in total, and since in the minimal surface each end-

point must be connected to only one other endpoint by a geodesic,

A consists of n geodesics.

We use the notation [n] ≡ {1, 2, . . . , n} as in [5].

The graph formed by the endpoints as vertices and the geodesics as edges is known as a complete

bipartite graph.

– 2 –

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确定AdS3/CFT2中Ryu-Takayanagi表面的算法与复杂性分析

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在全息共形场理论框架下，如何应用Ryu-Takayanagi公式以及量子校正和JLMS公式来探究纠缠熵与引力几何形状之间的关系？

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