Island Finder and Entropy Bound

TL;DR

This paper provides a practical framework to certify the existence of entanglement islands in semiclassical gravity using Wall's maximin construction and the Quantum Focusing Conjecture, deriving an upper bound on island-entropy that yields the Page-curve behavior in evaporating black holes across dimensions. The core idea is to identify simple sufficient conditions where an auxiliary region I' reduces the generalized entropy when joined with a reference system R, guaranteeing a nonempty island I with S_gen(I∪R)≤S_gen(I'∪R). The authors also prove a complementary entropy bound: in a globally pure state, the true entropy of R cannot exceed the generalized entropy of a suitable asymptotic region (external or distant), with explicit bounds expressed via S_gen on regions like I'_c. The approach is illustrated through multiple examples (evaporating BH after Page time, recollapsing FRW, bag-of-gold, collapsing star) and provides a general tool for understanding unitarity and information flow in spacetimes beyond AdS/CFT.

Abstract

Identifying an entanglement island requires exquisite control over the entropy of quantum fields, which is available only in toy models. Here we present a set of sufficient conditions that guarantee the existence of an island and place an upper bound on the entropy computed by the island rule. This is enough to derive the main features of the Page curve for an evaporating black hole in any spacetime dimension. Our argument makes use of Wall's maximin formulation and the Quantum Focusing Conjecture. As a corollary, we derive a novel entropy bound.

Figures (9)

• Figure 1: Left: evaporating black hole; right: its Page curve. After the Page time, the semiclassical entropy $S(R)$ of the Hawking radiation in the asymptotic region $R$ exceeds the Bekenstein-Hawking entropy of the black hole, $A_h/4G\hbar$. The "Hawking partners" in the black hole interior purify $R$. (Dashed lines indicate entanglement.) Therefore, adjoining $\hat{I}$ to $R$ decreases the generalized entropy $S_{\rm gen}$. However, islands must have stationary $S_{\rm gen}$. Solving for this condition exceeds present analytic control over the entropy. The island finder presented here sidesteps this obstruction.
• Figure 2: Island finder. Suppose that $I'\cup R$ is quantum normal (top) or anti-normal (bottom). Then the generalized entropy of $I'\cup R$ decreases along the dashed lines to $\tilde{I}'\subset \Sigma$. An island $I$ with even smaller generalized entropy $S_{\rm gen}(I\cup R)$ must exist on the maximin Cauchy slice $\Sigma$. If $S_{\rm gen}(I'\cup R)<S(R)$, the island cannot be empty.
• Figure 3: Maximin restricted to the domain of dependence ("wedge") $D(I_0)$ returns a region $\hat{I}$ on a maximin slice $\Sigma$. If $I_0 \cup R$ is quantum normal, then $\partial\hat{I}$ cannot intersect $\partial D(I_0)$ (dashed). Left: if $\hat{I}\cap \partial I_0\neq\varnothing$, then $S_{\rm gen}(\hat{I}(\epsilon)\cup R) < S_{\rm gen}(\hat{I}\cup R)$, contradicting the min of maximin. Right: if $\hat{I}\cap \partial D(I_0)-\partial I_0$ then $S_{\rm gen}(\hat{I}(\epsilon)\cup R) > S_{\rm gen}(\hat{I}\cup R)$ on the deformed slice $\Sigma(\epsilon)$ violates the max of maximin.
• Figure 4: Evaporating black hole after the Page time. Hawking radiation has accumulated in $R$. As shown on the left, the boundary of the past of $R$, denoted by $r_0(v)$, intersects the stretched horizon (shown in purple) at the sphere $A_s$, which together with the $A_h$ sphere on the event horizons reside at retarded time $v_s$. We consider candidate regions $I'$ with boundary $\partial I'$ on a causal horizon spacelike to $R$ (grey regions). The generalized second law implies that $S_{\rm gen}(I'\cup R)$ increases under future outward deformations. For future inward deformations, quantum normalcy follows from the (trivial) classical normalcy in the dark grey subregion, which is chosen to keep quantum corrections to the expansion small. The $I'$ that minimizes $S_{\rm gen}(I'\cup R)$ subject to these restrictions is shown in pink. Its boundary is located a $\Delta v = r_s \log c_1$ to the future of $A_h$, as shown on the right. We show that it provides an extremely tight upper bound on the true entropy $S(\mathbf{R})=S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I' \cup R) = A_h /4G\hbar + O(1)$.
• Figure 5: Spatially flat radiation-dominated universe with negative cosmological constant, purified by a reference universe (thermal Minkowski space, right). If we choose a large enough reference region $R$ at $t_{\rm Mink}=0$, then the region $I'$ at the turnaround time $t=0$ satisfies our sufficient conditions. Therefore an island $I$ must exist.