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Real-space RG, error correction and Petz map

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arxiv 2012.14001 v4 pith:X2IPJJNQ submitted 2020-12-27 hep-th math-phmath.MPquant-ph

Keiichiro Furuya , Nima Lashkari , Shoy Ouseph This is my paper
classification hep-thmath-phmath.MPquant-ph
keywords erroroperatorscorrectionalgebraquantumreal-spacerecoverycomplementary
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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