Quantum complexity in the AdS/CFT correspondence

AdS/CFT对应，通常被称为全息，展示了引力理论和量子场论（QFT）之间的显着二重性。最近，在Ryu和Takayanagi（2006）提出全息纠缠熵之后，引力理论和量子信息之间建立了新的关系。特别是，为了获得对黑洞量子性质的新见解，Susskind和合作者提出了量子信息概念（如计算复杂性）的引力实现。然而，到目前为止，全息复杂性理论和量子信息复杂性之间的明确关系仍然是一个悬而未决的问题。进一步的进展需要将信息论定义推广到QFT，即无限维希尔伯特空间。最近，免费QFT在这个方向上取得了进展。在这里，作为一个有前途的路径来理解相互作用的QFT的复杂性，我们将提出和分析复杂性的定义在二维共形场理论（CFTs），CFTs和没有AdS对偶。具体地说，我们的工作包括以下三个部分及其相互关系：1）在AdS/CFT中，我们将利用Wilson线确定三维渐近AdS空间中非极小测地线的场论性质。预计这些量子点是激发态的CFT两点函数。使用运动空间，从积分几何的数学概念，我们将提供非最小测地线的全息复杂性的建议的贡献的场论解释。2)在CFT的背景下，我们计划扩展Caputa等人最近提出的复杂性建议，该建议涉及Virasoro代数的单个表示，以更一般的情况下，Kac-Moody对称代数。我们将显式地计算不同参考和目标状态的复杂度。此外，我们计划把这个建议的一个中心元素，1+1维Polyakov作用量，与AdS_3引力联系起来。3)从具有已知晶格正则化（如伊辛模型）的CFTs的例子开始，我们将研究哪些门变换，参考状态和复杂性度量与连续极限相容，从而与共形对称性相容。我们将使用这些结果来计算Virasoro代数的不同表示中的状态的复杂性，作为第2部分的推广。最后，我们将联合收割机的建议，在有助于澄清CFT和全息内的复杂性建议之间的关系的三个部分的结果。我们希望我们的研究结果能为研究黑洞的量子性质提供更多的新途径。

The AdS/CFT correspondence, often referred to as holography, conjectures a remarkable duality between gravity theories and quantum field theories (QFT). Recently, following the holographic entanglement entropy proposal of Ryu and Takayanagi (2006), new relations between gravity theories and quantum information have been established. In particular, with the aim of gaining new insight into the quantum nature of black holes, gravity realizations of quantum information concepts such as computational complexity were proposed by Susskind and collaborators. So far, however, the explicit relation between holographic complexity proposals and the complexity of quantum information remains an open question. Further progress requires a generalization of the information-theoretic definition to QFTs, i.e. to infinite-dimensional Hilbert spaces. Recently, progress in this direction was made for free QFTs. Here, as a promising path to understanding complexity for interacting QFTs, we will propose and analyze complexity definitions in two-dimensional conformal field theories (CFTs), both for CFTs with and without AdS dual. In particular, our project consists of the following three parts and their interrelation: 1) Within AdS/CFT, we will determine the field theory duals of non-minimal geodesics in three-dimensional asymptotically AdS spaces by means of Wilson lines. These duals are expected to be CFT two-point functions for excited states. Using kinematic space, a mathematical concept from integral geometry, we will provide afield-theory interpretation of the contribution of non-minimal geodesics to proposals for holographic complexity. 2) In the context of CFT, we plan to extend a recent complexity proposal of Caputa et al that involves a single representation of the Virasoro algebra to the more general case of Kac-Moody symmetry algebras. We will explicitly calculate complexity as given by this proposal for different reference and target states. Moreover, we plan to relate a central element of this proposal, the 1+1-dimensional Polyakov action, to AdS_3 gravity. 3) Starting from examples for CFTs with known lattice regularizations such as the Ising model, we will investigate which gate transformations, reference states and complexity measures are compatible with the continuum limit and thus with conformal symmetry. We will use these results to compute complexity for states in different representations of the Virasoro algebra, in generalization of part 2). Finally, we will combine the results of the three parts of the proposal in view of contributing to clarifying the relation between complexity proposals within CFT and holography. We expect our results to indicate further new avenues for studying aspects of the quantum nature of black holes.