Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy

TL;DR

The paper develops a covariant framework for holographic entanglement entropy using the maximin construction, proving that maximin surfaces coincide with HRT extremal surfaces under the null curvature condition. It then shows these surfaces have favorable geometric properties: they lie outside the causal wedge, move outward as boundary regions grow, and satisfy strong subadditivity and monogamy of mutual information. Existence is established for horizonless spacetimes and for black holes with Kasner-like singularities, with a stability analysis ensuring at least one stable maximin surface. The results bolster the validity of the covariant holographic entropy proposal and provide a robust toolkit for bulk reconstruction and information-theoretic inequalities, while outlining directions to incorporate stringy and semiclassical corrections.

Abstract

The covariant holographic entropy conjecture of AdS/CFT relates the entropy of a boundary region R to the area of an extremal surface in the bulk spacetime. This extremal surface can be obtained by a maximin construction, allowing many new results to be proven. On manifolds obeying the null curvature condition, these extremal surfaces: i) always lie outside the causal wedge of R, ii) have less area than the bifurcation surface of the causal wedge, iii) move away from the boundary as R grows, and iv) obey strong subadditivity and monogamy of mutual information. These results suggest that the information in R allows the bulk to be reconstructed all the way up to the extremal area surface. The maximin surfaces are shown to exist on spacetimes without horizons, and on black hole spacetimes with Kasner-like singularities.

Figures (3)

• Figure 1: The picture proof of Strong Subadditivity for the Ryu-Takayanagi conjecture on a static slice HT07. The horizontal line represents the boundary, the solid lines are the surfaces $\mathrm{min}(AB)$ and $\mathrm{min}(BC)$, and the dashed lines are the surfaces $\mathrm{min}(ABC)$ and $\mathrm{min}(B)$. The area of $\mathrm{min}(B)$ is less than $de$, while the area of $\mathrm{min}(ABC)$ is less than $fg$. (Note that the difference between these areas cannot diverge at the boundary; otherwise it would be more area-efficient for $\mathrm{min}(AB)$ and $\mathrm{min}(BC)$ to coincide exactly with $\mathrm{min}(ABC)$ and $\mathrm{min}(B)$ in a neighborhood of the boundary.)
• Figure 6: Examples of unstable maximin surfaces: (i) An eternal static AdS-Schwarzschild black hole. The bifurcation surface $x$ is a maximin surface, and is also extremal. However, technically, any other slice $y$ of the past or future horizon is also a maximin surface, because there exist achronal slices $\Sigma$ which follow the horizon, and $\mathrm{Area}(x) = \mathrm{Area}(y)$. These other slices $y$ are unstable because if $\Sigma$ is slightly deformed (as shown by the dotted line), there is no longer a minimal area surface near $y$. These unstable maximin surfaces disappear when the horizons are not exactly static. (ii) A wormhole with two extremal surfaces $x$, $z$ in its throat, such that $\mathrm{Area}(z) > \mathrm{Area}(x) = \mathrm{Area}(y)$. Once again, $x$ is the stable maximin surface, but $y$ is an unstable surface since $\Sigma$ can be chosen to pass through $y$. In neither case (i) nor (ii) is $y$ extremal.
• Figure 9: $H(A)$ is the intersection of the past and future horizon continued beyond the causal surface $w(A)$. Because the slice on which $M(A)$ is minimal intersects $H(A)$ at $\tilde{w}$, by the Second Law, $M(A)$ has less area than $w(A)$.