Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy (1211.3494v4)

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.

• The paper establishes that maximin surfaces satisfy strong subadditivity and are equivalent to HRT surfaces in the AdS/CFT framework.
• The paper employs a maximin procedure to rigorously prove that extremal surfaces extend beyond causal wedges even in spacetimes with black holes and singularities.
• The paper provides a theoretical framework that enhances the geometric understanding of quantum entanglement and guides future research on quantum gravity corrections.

Holographic Entanglement Entropy and Maximin Surfaces

The paper "Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy" by Aron C. Wall presents a detailed exploration of the relationship between covariant holographic entropy and maximin surfaces within the AdS/CFT framework. This work advances the theoretical understanding of how entropic quantities in conformal field theories (CFTs) relate to geometric structures in corresponding anti-de Sitter (AdS) spacetimes.

Key Contributions

The central premise of this paper is the exploration of extremal surfaces obtained through a maximin procedure, which allows the proof of several significant results related to holographic entanglement entropy:

1. Extremal Surfaces Characteristics: The extremal surfaces stay outside the causal wedge boundary as the boundary region increases in size. These surfaces, attained from a maximin construction, comply with strong subadditivity and exhibit monogamy of mutual information.
2. Existence in Various Spacetimes: The existence of maximin surfaces extends beyond horizonless spacetimes to those containing black holes with Kasner-like singularities, indicating their broad applicability in diverse spacetime geometries.
3. Maximin-HRT Equivalence: The paper demonstrates that maximin surfaces are equivalent to Hubeny-Rangamani-Takayanagi (HRT) surfaces, thus ensuring that the properties and proofs developed for maximin surfaces directly apply to HRT surfaces.
4. Properties: The paper establishes that maximin surfaces possess several desirable properties: they move outwards as the boundary region grows and possess less area than the causal surface, foundational traits in the holographic context.
5. Strong Subadditivity and Monogamy: By employing the maximin strategy, the paper comprehensively proves the strong subadditivity of holographic entanglement entropy in dynamic spacetimes. It further extends these methods to verify the monogamy of mutual information.

Implications and Future Directions

The implications of these findings are profound both for theoretical physics and potential computational uses in understanding quantum gravity contexts:

• Reconstruction of Bulk Geometry: The maximin construction provides insights into how information from a boundary region can stretch over a bulk, hinting at robust geometrical underpinnings of holographic theories.
• Theoretical Foundation: The rigorous proofs of entropy inequalities strengthen the theoretical foundation of AdS/CFT correspondence, highlighting its capability to encapsulate quantum information theoretic insights through geometric realizations.
• Extensions Beyond Classical GR: While these results were derived within classical GR, they suggest directions for extending or modifying the theory when considering stringy or semiclassical corrections, such as in Lovelock gravity or with entanglement entropy perturbations.

Potential Research Pathways

Advancing beyond the classical scope, one significant frontier is exploring the impacts of string-theoretic or semiclassical corrections, especially those altering the null curvature condition (NCC), on the formulation and implications of holographic entanglement entropy. Furthermore, understanding the behavior of these geometric constructs in scenarios that include dynamical spacetimes or complex singularity structures, such as those involving inflationary cosmologies or de Sitter boundaries, remains an open and compelling research avenue.

In summary, Wall's work provides a robust and formal framework for understanding entanglement entropy in dynamical settings and sets the stage for future investigations into the subtleties of holography and quantum gravity theory.