A boundary region's infinite-N operator algebra is a von Neumann algebra exactly when its generalized causal wedge closes on the same region; null geodesic focusing is the bulk mechanism.
No Simple Dual to the Causal Holographic Information?
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In AdS/CFT, the fine grained entropy of a boundary region is dual to the area of an extremal surface X in the bulk. It has been proposed that the area of a certain 'causal surface' C - i.e. the 'causal holographic information' (CHI) - corresponds to some coarse-grained entropy in the boundary theory. We construct two kinds of counterexamples that rule out various possible duals, using (1) vacuum rigidity and (2) thermal quenches. This includes the 'one-point entropy' proposed by Kelly and Wall, and a large class of related procedures. Also, any coarse-graining that fixes the geometry of the bulk 'causal wedge' bounded by C, fails to reproduce CHI. This is in sharp contrast to the holographic entanglement entropy, where the area of the extremal surface X measures the same information that is found in the 'entanglement wedge' bounded by X.
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The Making of von Neumann Algebras from Bulk Focusing
A boundary region's infinite-N operator algebra is a von Neumann algebra exactly when its generalized causal wedge closes on the same region; null geodesic focusing is the bulk mechanism.