Real-space RG, error correction and Petz map

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.

Snowmass White Paper: Quantum Aspects of Black Holes and the Emergence of Spacetime

“…Further applications of modular flow to these issues can be found in[53,55,[111][112][113][114]]. 7 See for example[88,91,92,[115][116][117][118].…”

Renormalization Group Flow as Optimal Transport

“…More broadly, it seems that many more tools from optimal transport, possibly combined with information theory, can be brought to bear on the subject of ERG via our present formulation. For instance, it appears likely that our formulation of RG flow in this manuscript could be synthesized with the approaches of [34][35][36][37][38][39] from the physics community. Since the optimal transport community has enormous analytical and numerical traction in the PDE setting, it would be valuable to adapt these insights to the functional generalizations appropriate for ERG flows.…”

“…For instance, the relative entropy quantifies how much additional memory is required to compress a list of samples from P if we are only given just enough memory to optimally compress a list of as many samples from Q. There have been other works on RG flow which have leveraged the relative entropy [34][35][36][37][38][39], albeit in a manner which does not involve optimal transport.…”

The Universe as a Quantum Encoder

“…Moreover it is understood how inverse RG flow gives rise to isometries in the language of tensor networks (see e.g. [38][39][40][41][42][43][44][45][46][47][48][49][50][51]) and quantum error correction [17][18][19]. Hence we propose here as a refinement of dS/CFT that the time evolution of states is given by isometries, and that these isometries are generated by inverse RG flow of the dual field theory at I + .…”

“…Our third approach leverages beautiful work on AdS 3 /CFT 2 holography [4][5][6], where radial evolution at fixed time in AdS 3 was modeled by a tensor network quantum error-correcting code. This is intimately tied the fact that radial evolution is inverse RG flow, which is known in general to be an approximate error-correcting code [17][18][19]. This work is combined with the construction of quantum gravity in de Sitter space as a braneworld near the boundary of AdS [20].…”