# No Simple Dual to the Causal Holographic Information?

## TL;DR

This paper challenges the proposed duality between causal holographic information and boundary coarse-grained entropy in AdS/CFT, providing counterexamples that show the duality does not hold in general.

## Contribution

The authors construct counterexamples using vacuum rigidity and thermal quenches to disprove various proposed duals of causal holographic information.

## Key findings

- Counterexamples invalidate several dual proposals for CHI.
- Fixing the bulk causal wedge geometry does not reproduce CHI.
- Holographic entanglement entropy differs fundamentally from CHI in duality properties.

## Abstract

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.