On infinite tensor networks, complementary recovery and type II factors

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 Harlow-Pastawski-Preskill-Yoshida 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 ∞ 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 multiscale entanglement renormalization ansatz 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 ansätze for approximating subregions of quantum field theories.

We thank Charles Cao and Zoltán Zimborás for numerous helpful initial insights and comments. We also thank Shadi Ali Ahmad, Charlie Cummings, Jens Eisert, and Alexander Stottmeister, for illuminating discussions and comments on the draft. L S thanks Lorenzo Leone, Lennart Bittel and Dimitris Saraidaris for discussions in the course of related work. The authors would like to express their sincere gratitude for Wayne Weng’s early-stage support and commitment to this project, which was essential to its development. A J and L S are grateful for support from the Einstein Foundation Berlin and the Einstein Research Unit ‘Perspectives of a quantum digital transformation’. E G is supported by the Office of High Energy Physics of U.S. Department of Energy under Grant Contract Number DE-SC0012567 and DE-SC0020360 (MIT Contract Number 578218). D M is supported by the Office of High Energy Physics of the U.S. Department of Energy under Grant Contract Number DE-SC0018407.