Quantum complexity in the AdS/CFT correspondence

AdS/CFT对应，通常被称为全息，推测了引力理论和量子场论（QFT）之间的显著对偶性。近年来，继Ryu和Takayanagi（2006）提出全息纠缠熵之后，引力理论与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本质的新见解，Susskind及其合作者提出了量子信息概念（如计算复杂性）的重力实现。然而，到目前为止，全息复杂性建议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的定义推广到量子傅里叶变换，即推广到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，作为理解相互作用qft复杂性的一个有希望的途径，我们将提出并分析二维共形场理论（cft）中的复杂性定义，包括具有和不具有AdS对偶的cft。具体而言，我们的项目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对偶。这些对偶被期望为激发态的CFT两点函数。利用运动空间，一个来自积分几何的数学概念，我们将提供非极小测地线对全息复杂性建议的贡献的场论解释。2)在CFT的背景下，我们计划将Caputa等人最近提出的涉及Virasoro代数的单一表示的复杂性建议扩展到更一般的Kac-Moody对称代数的情况。对于不同的参考状态和目标状态，我们将显式地计算该建议给出的复杂度。此外，我们计划将该提议的一个中心元素，即1+1维Polyakov作用，与AdS_3引力联系起来。3)从具有已知晶格正则化（如Ising模型）的cft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统极限兼容，从而与共形对称兼容。在第2部分的推广中，我们将使用这些结果来计算不同Virasoro代数表示中的状态的复杂性。最后，我们将把提案的三个部分的结果结合起来，以有助于澄清CFT和全息术中复杂性提案之间的关系。我们希望我们的结果能够为研究黑洞的量子性质提供进一步的新途径。

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.