Real-space renormalization, error correction, and conditional expectations

Speaker:Nima Lashkari
Time: Mon. 11th January 10:00
Venue:Zoom ID: 828 1741 2509 Password: 940025


We show that the real-space renormalization group (RG), as a map from the observable algebra to the subalgebra of long-distance observables, is an error correction code, best described by a conditional expectation. It is comprised of a coarse-graining step followed by an isometric embedding. The coarse-graining is the error map and the long-distance observables are the correctable operators. We show that if there is a state that is preserved under renormalization the coarse-graining step is the Petz dual of the isometric embedding (the Petz map). We demonstrate that a set of states are preserved under this map if and only if their pairwise relative entropies do not change when we restrict to the long-distance observables. We study the operator algebra quantum error correction in the GNS Hilbert space which applies to any quantum system including the local algebra of quantum field theory. We show that the recovery map is an isometric embedding of the correctable subalgebra. Similar to the RG, the composition of the error map followed by the recovery map forms a conditional expectation (a projection in the GNS Hilbert space). In gauge/gravity dualities, the bulk relative entropy of holographic states is the same as their boundary relative entropies which implies that the holographic map is an error correction code, and hence a conditional expectation. It follows that the boundary to the bulk map is a Petz map.
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