Quantum Maximin Surfaces

TL;DR

This paper develops a quantum analogue of Wall's maximin construction, proving that quantum maximin surfaces are equivalent to minimal quantum extremal surfaces and extending the framework to hybrid entropies that include nonholographic subsystems (islands). It carefully analyzes subtleties from UV divergences, corners, and boundary conditions, and establishes foundational consistency checks by proving entanglement wedge nesting and strong subadditivity, even in the presence of perturbative bulk quantum backreaction. The results further generalize to nonholographic subsystems, showing that hybrid entropy extremization obeys nesting and SSA, thereby supporting island-type proposals in AdS/CFT. Collectively, the work provides a robust, operational toolkit for studying quantum-corrected holographic entanglement and its extensions in evaporating black holes and related setups.

Abstract

We formulate a quantum generalization of maximin surfaces and show that a quantum maximin surface is identical to the minimal quantum extremal surface, introduced in the EW prescription. We discuss various subtleties and complications associated to a maximinimization of the bulk von Neumann entropy due to corners and unboundedness and present arguments that nonetheless a maximinimization of the UV-finite generalized entropy should be well-defined. We give the first general proof that the EW prescription satisfies entanglement wedge nesting and the strong subadditivity inequality. In addition, we apply the quantum maximin technology to prove that recently proposed generalizations of the EW prescription to nonholographic subsystems (including the so-called "quantum extremal islands") also satisfy entanglement wedge nesting and strong subadditivity. Our results hold in the regime where backreaction of bulk quantum fields can be treated perturbatively in $G_{N}\hbar$, but we emphasize that they are valid even when gradients of the bulk entropy are of the same order as variations in the area, a regime recently investigated in new models of black hole evaporation in AdS/CFT.

Figures (9)

• Figure 1: The quantum extremal surface ${\cal X}_{R}$ for boundary subregion $R$ with associated entanglement wedge $W_{E}[R]$. The dashed lines are spacelike as a result of caustics, which occur generically.
• Figure 2: The surfaces $\sigma$ and $\sigma'$ are shown tangent at the point $p$. The arrows illustrate the projection of the null orthogonal vectors onto the Cauchy slice. The associated null expansion of $\sigma'$ lowerbounds that of $\sigma$.
• Figure 3: We consider three types of boundary conditions: a) reflecting, b) absorbing, and c) open. For absorbing and open boundary conditions, the choice of the Cauchy slice for evaluating the von Neumann entropy is no longer immaterial, as degrees of freedom can enter and exit the bulk.
• Figure 4: Two timelike-separated regions of the boundary are depicted, $R$ and $R'$, with $R' \subset D^+[R]$. The boundary conditions allow modes to pass from the bulk, through the boundary, into the system $E$. The bulk surface $M$ is anchored to $\partial R = \partial R'$ and timelike related to $R$ but spacelike related to $R'$. While $\sigma$ does not lie on any Cauchy slice anchored to $R$, we can nevertheless define an entropy of $\sigma$ with respect to $R$ as the entropy of $H_{R'}\cup E$.
• Figure 5: We define the representatives of a surface $\sigma$ on Cauchy slice $C$ as the intersection of $C$ with the boundary of $J[\sigma]\equiv J^{+}[\sigma]\cup J^{-}[\sigma]$. We display the case where there are two representatives of $\sigma$ on $C$.

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• proof
• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof