标题： 关于无限张量网络、互补恢复和II型因子 显示英文标题

标题： On Infinite Tensor Networks, Complementary Recovery and Type II Factors

摘要： 我们开始研究无限张量网络边界处的局部算子代数，使用归纳极限的数学理论。 特别是，我们考虑了每层都作为具有互补恢复量子码的张量网络，这一性质在全息量子纠错码的边界到体映射中起着突出作用。 在这种情况下，我们将限制Hilbert空间和可观测量代数分解，以便跟踪网络中的纠缠。 作为一个具体例子，我们描述了全息HaPPY码模型的归纳极限，并将其代数和纠错特性联系起来。 我们发现该模型中的局部代数由超有限型II$_\infty$因子给出。 接下来，我们讨论了基于此框架构建的其他网络，并评论了II型因子与稳定器电路之间的联系。 最后，我们讨论了互补恢复被破坏的MERA网络。我们认为这种破坏可能允许一个限制性的III型冯·诺依曼代数，使它们更适合用作近似量子场论子区域的候选者。 显示英文摘要

摘要： We initiate a study of local operator algebras at the boundary of infinite tensor networks, using the mathematical theory of inductive limits. In particular, we consider tensor networks in which each layer acts as a quantum code with complementary recovery, a property that features prominently in the bulk-to-boundary maps intrinsic to holographic quantum error-correcting codes. In this case, we decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network. As a specific example, we describe this inductive limit for the holographic HaPPY code model and relate its algebraic and error-correction features. We find that the local algebras in this model are given by the hyperfinite type II$_\infty$ factor. Next, we discuss other networks that build upon this framework and comment on a connection between type II factors and stabilizer circuits. We conclude with a discussion of MERA networks in which complementary recovery is broken. We argue that this breaking possibly permits a limiting type III von Neumann algebra, making them more suitable ans\"atze for approximating subregions of quantum field theories.