[论文解读] Asymptotically isometric codes for holography

Thomas Faulkner, Min Li|arXiv (Cornell University)|Nov 22, 2022

Cosmology and Gravitation Theories被引用 10

一句话总结

本文提出了全息理论中的渐近等距量子误差纠正码，尽管在有限N时嵌入并非等距，但在大N极限下，体量子场论代数仍能涌现。该文证明了通过网扩张，因果楔形和纠缠楔形重建得以保留，建立了渐近信息扰动权衡定理，并表明纠缠楔形扩张存在当且仅当体与边界相对熵或模流在渐近意义上一致。

ABSTRACT

The holographic principle suggests that the low energy effective field theory of gravity, as used to describe perturbative quantum fields about some background has far too many states. It is then natural that any quantum error correcting code with such a quantum field theory as the code subspace is not isometric. We discuss how this framework can naturally arise in an algebraic QFT treatment of a family of CFT with a large-$N$ limit described by the single trace sector. We show that an isometric code can be recovered in the $N ightarrow \infty$ limit when acting on fixed states in the code Hilbert space. Asymptotically isometric codes come equipped with the notion of simple operators and nets of causal wedges. While the causal wedges are additive, they need not satisfy Haag duality, thus leading to the possibility of non-trivial entanglement wedge reconstructions. Codes with complementary recovery are defined as having extensions to Haag dual nets, where entanglement wedges are well defined for all causal boundary regions. We prove an asymptotic version of the information disturbance trade-off theorem and use this to show that boundary theory causality is maintained by net extensions. We give a characterization of the existence of an entanglement wedge extension via the asymptotic equality of bulk and boundary relative entropy or modular flow. While these codes are asymptotically exact, at fixed $N$ they can have large errors on states that do not survive the large-$N$ limit. This allows us to fix well known issues that arise when modeling gravity as an exact codes, while maintaining the nice features expected of gravity, including, among other things, the emergence of non-trivial von Neumann algebras of various types.

研究动机与目标

• 解决全息理论中局部量子场论代数与精确量子误差纠正码之间的不相容性问题。
• 解决有限-N有效场论过度计数自由度的问题，导致非等距嵌入。
• 将QEC框架从因果楔形扩展至纠缠楔形，利用大-N CFT。
• 建立在近似码中实现非平凡纠缠楔形重建的条件。
• 证明渐近版本的信息扰动权衡定理，以保持边界因果性。

提出的方法

• 从大-N CFT中的单迹算符构造码子空间，其在N→∞极限下生成体有效场论。
• 使用有界算符VN将码嵌入微观理论，允许非平凡核与有限N下的非等距编码。
• 通过编码映射在大-N极限下对固定态收敛于等距算符，定义渐近等距码。
• 引入对Haag对偶网的网扩张，以定义行为良好的纠缠楔形并实现非平凡重建。
• 应用Borchers的类时管定理与格林函数论证，以证明因果域的可加性及因果楔形对偶性。
• 通过体与边界相对熵或模流的渐近相等性，建立纠缠楔形扩张的判据。

实验结果

研究问题

• RQ1 在全息理论中，非等距码是否仍能支持一致的因果楔形与纠缠楔形重建？
• RQ2 尽管码子空间为有限维，体量子场论代数（特别是III1型冯诺依曼代数）如何在QEC框架中涌现？
• RQ3 在具有有限-N误差的近似码中，何种条件可确保纠缠楔形重建的实现？
• RQ4 当码在有限N下非等距时，边界因果性如何得以保持？
• RQ5 纠缠楔形扩张存在的条件是什么，以及如何对其进行表征？

主要发现

• 渐近等距码在N→∞极限下对固定态恢复等距行为，解决了有限-N QFT中自由度过度计数的问题。
• 纠缠楔形扩张的存在性等价于体与边界相对熵或模流的渐近相等性。
• 对Haag对偶网的网扩张使得纠缠楔形定义良好，并实现非平凡重建，即使因果楔形不满足Haag对偶性。
• 证明了渐近信息扰动权衡定理，确保在码扩展下边界因果性得以保持。
• Borchers的类时管定理与格林函数密度论证支持因果域的可加性，为体中因果楔形对偶性提供依据。
• 该框架允许非平凡III1型冯诺依曼代数的涌现，与QFT真空纠缠一致，同时与全息界限相容。