Effective entropy of quantum fields coupled with gravity

TL;DR

The paper develops an effective entropy for quantum fields in dynamical spacetimes by extending the replica trick to include gravity, introducing a brane mechanism that stabilizes replica geometries. The resulting quantum extremal surface formula generalizes holographic entropy to general geometries without AdS boundaries and naturally yields entanglement islands when matter is highly entangled, addressing questions like the Page curve in evaporating black holes. It applies the framework to a 2D gravity+QFT density-matrix example and a 4D Schwarzschild spacetime, illustrating Page-like transitions and island formation. To deepen intuition in non-AdS settings, the authors develop random tensor network models and a super-density operator formalism, showing how islands can be realized and information recovered in a region-dependent, observer-dependent manner. The work also discusses the relation between closed- and open-universe pictures and outlines future directions for nonperturbative definitions and observer-aware reconstruction in dynamical geometries.

Abstract

Entanglement entropy quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a gravitational path integral on replica geometry with a co-dimension-$2$ brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. By introducing ancilla systems, we show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.

Figures (22)

• Figure 1: Illustration of the QES formula in Eq. (\ref{['eq:QES']}) for two different situations. The curved black lines represent a Cauchy surface. (a) $\Sigma=A$ when the formula reduced to the ordinary quantum field theory entropy. (b) $\Sigma=A\cup I$ when a new quantum extremal surface $\gamma=\partial I$ appears, contributing an area law entropy $\frac{|\gamma|}{4G_N}$. The quantum field theory entropy becomes $S^{\rm qft}_{AI}$ instead of $S^{\rm qft}_{A}$, which can reduce the entropy when there is entanglement between $I$ and $A$, as is indicated by the red dashed lines.
• Figure 2: Illustration of the replica calculation of Renyi entropy in fixed background (Eq. (\ref{['eq:QFTentropy']})) for $n=2$. The replica geometry has conical singularity at the branch surfaces.
• Figure 3: The left panel is the original black hole geometry, with a bulk region defined either by sending two light rays from the boundary (the orange lines) or by fixing the proper distance from the boundary (the blue line). The right panel is the two replica wormhole geometry, which has a temperature lower than the original black hole temperature. The dashed curve is the original boundary and the solid curve is that in the two replica geometry. Using the same light rays will thus define a bulk region that is larger than the region defined by fixing proper distance.
• Figure 4: The $n$-replica geometry $\mathcal{M}_n$ with a possible replica wormhole (left), and the $Z_n$ quotient geometry $\mathcal{M}_n/Z_n$. The quotient geometry has no conical singularity at the boundary of $A$, but has a conical singularity with angle $\frac{2\pi}{n}$ at the boundary of the extra branching surface ( i.e. the "island") $I$.
• Figure 5: The microscopic (left panel) andgGravitational (right panel) description of the CFT density matrix.