Real-space RG, error correction and Petz map

TL;DR

This paper explores the error correction capabilities of real-space RG in quantum systems, compares it to holography, and extends operator algebra quantum error correction to von Neumann algebras, highlighting conditions for recovery maps.

Contribution

It analyzes the error correction properties of real-space RG and extends operator algebra quantum error correction to von Neumann algebras with finite index conditions.

Findings

Long-distance operators are approximately correctable in real-space RG.

Real-space RG lacks the complementary recovery property unlike holography.

Petz dual of an error map serves as a recovery map under finite index inclusion.

Abstract

There are two parts to this work: First, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large $N$ and a large gap in the spectrum of operators in the emergence of complementary recovery. Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.