III型冯·诺依曼代数是神奇的
Type III von Neumann Algebras are Magical
AI总结：
本研究证明III型冯·诺依曼代数根本需要无限量魔资源，有限维系统热力学极限下若态仅具有限魔资源，其局域冯·诺依曼代数不可能为III型，结果对量子场论量子模拟有直接意义。
AI中文摘要：
执行任务所需的非Clifford门数量，或简称“magic”（魔资源），是容错量子计算的一种资源。冯·诺依曼代数为描述无限维量子系统提供了形式化数学结构，这类系统包括量子场论或量子统计力学中的系统。一类特别重要的冯·诺依曼代数被称为III型代数，对于这类代数，有限维系统的标准概念如密度矩阵和迹会失效。在本研究中，我们提出III型冯·诺依曼代数从根本上需要无限量的魔资源。具体而言，我们考虑有限维量子系统的热力学极限，比如格点系统，并证明如果热力学极限下的态仅具有有限量的魔资源，那么得到的局域冯·诺依曼代数不可能是III型的。我们的结果对量子场论的量子模拟有直接意义，已知量子场论中局域子区域的代数是III型的。
英文摘要：
The number of non-Clifford gates needed to perform a task, or simply \textit{magic}, is a resource for fault-tolerant quantum computation. Von Neumann algebras provide a formal mathematical structure to describe infinite-dimensional quantum systems, such as those in quantum field theory or quantum statistical mechanics. A particularly important class of von Neumann algebras is called Type III algebras, for which the standard notions of finite-dimensional systems such as density matrices and traces break down. In this work, we argue that Type III von Neumann algebras fundamentally require an infinite amount of magic. Specifically, we consider the thermodynamic limit of finite-dimensional quantum systems, such as lattice systems, and show that if the states in the thermodynamic limit possess only a bounded amount of magic, the resulting local von Neumann algebra cannot be of Type III. Our result has direct implications for the quantum simulations of quantum field theories, for which the algebra of a local subregion is known to be of Type III.
