Quantum maximin surfaces

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 , 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.

Including contributions from entanglement islands to the reflected entropy

“…In the presence of gravitating regions, the maximin formula for computing entanglement entropy is given by [12]…”

“…Consider the subregion A 00 of Σ A∶B that minimizes SðA; A 00 Þ Σ A∶B , i.e., the candidate minimal entanglement island on Σ A∶B . Now, since A 00 ⊂ IsðABÞ using nesting, we can define B 00 ¼ IsðABÞnA 00 [12,38]. Then, from the maximin procedure, we have…”

“…Consider the Cauchy slice Σ, on which each of A ∪ IsðAÞ, B ∪ IsðBÞ and AB ∪ IsðABÞ are extremal, which exist by the nesting arguments made in [12,38]. Σ ∩ DðlsðABÞÞ, i.e., the island portion of the Cauchy slice, can be partitioned into A 0 and B 0 , which minimizes S R ðA; A 0 ∶B; B 0 Þ Σ .…”

“…This simplifies the analysis, since we can simply do the entire calculation in flat space by an appropriate Weyl transformation of Eq. (12). Let us first review the salient aspects of computing S n ðρ m=2 AB Þ for a CFT 2 in flat space as described in [23].…”

“…In a general timedependent situation, the location of the island is computed using a maximin procedure as we review in Sec. II [12]. This formula was first justified by considering holographic matter which itself has a bulk dual in [8] and then proved using the gravitational path integral in [9,10].…”