No Simple Dual to the Causal Holographic Information?

TL;DR

The paper investigates whether the causal holographic information $\chi_{\mathcal{R}}$ can be dual to a simple boundary coarse-grained entropy. It presents two robust counterexamples: a rigid interior geometry where the causal wedge fixes the entanglement wedge yet $\chi_{\mathcal{R}}$ remains nonzero, and a thermal-quench construction where fixing a single one-point function yields $\mathcal{S}^{(1)}=S_{\text{bdy}}$ but $\chi_{\mathcal{R}}$ stays larger than the area of the HRT surface $X$. Extending the argument to quantum corrections via the generalized entropy $S_{\text{gen}}$ and the quantum extremal surface $\varkappa_{\mathcal{R}}$, the authors show the no-duality result persists beyond the classical limit. The results reveal a fundamental distinction between the causal and entanglement wedges and suggest CHI may not admit a simple information-theoretic dual within entropy-maximization frameworks.

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.

Figures (5)

• Figure 1: The causal wedge, given by the region in past and future causal contact with a boundary subregion. The causal surface $C$ is the intersection of the past and future horizons. The HRT surface X lies outside of the causal wedge and in a spacelike direction Wal12.
• Figure 2: The cut-and-paste geometry is constructed from six patches of different mass AdS-Schwarzschild. These regions are patched together via AdS-Vaidya in the thin-shell limit. The bifurcation surface $B$ is the causal surface of the entire boundary. It lies well within the vacuum AdS ($M=0$) region of the spacetime.
• Figure 3: (a) A patch of AdS (zero mass Schwarzschild, in more than 3 bulk dimensions. (b)(c) Patches of mass $M$ AdS-Schwarzschild. (d)(e)(f) Patches of mass $2M$ AdS-Schwarzschild.
• Figure 4: The two geometries, corresponding to an unquenched (AdS-Schwarzschild) geometry, and the quenched geometries, are superposed. The extremal surface $X$ is unaltered by the matter, while the causal surface $C$ is displaced. The areas in gray represent the region that is perturbed by matter, and the green lines represent its wavefront (whose maximum velocity is the speed of light). The Euclidean geometry is drawn in blue as an additional branch, coinciding with the Lorentzian geometry at $t=0$. (Because of the possibility of caustics, the null surfaces coming from $X$ and $C$ are drawn with different angles, even though both surfaces are generated by null geodesics.
• Figure 5: An illustration of all possible constraints for a coarse-grained entropy dual to CHI. We have ruled out a maximization coarse-graining procedure constrained by (1) the set of all boundary one-point functions, (2) any superset of the boundary one-point functions, (3) any subset of the one-point functions giving rise to well-defined states, (4) any superset of (3), and (5) any subset of the one-point functions giving rise to an ill-defined state. The remaining options are supersets of (5), or alternative coarse-graining procedures not based on entropy maximization.