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arXiv:2605.28958

Quantum State of a Gravitating Region

Raphael Bousso, Sami Kaya, Guanda Lin, Arvin Shahbazi-Moghaddam

hep-thgr-qcquant-phSubmitted May 27, 2026• Announced May 29, 2026
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Abstract

We propose that any compact $d$-manifold with elliptic data, $\mathcal{J}$, prepares a quantum state $|\mathcal{J}\rangle$ on its $(d-1)$-boundary $σ$. Elliptic data consists of metric and field values, or their conjugates, but not both. No asymptotic structure is required. Inner products and traces are evaluated by the gravitational path integral with closed boundary conditions obtained by gluing elliptic data manifolds. In particular, we give a prescription for the Rényi entropies $S_n$ of a subregion of $σ$. In a class of examples, we find that $S_n$ is nonnegative and nonincreasing with $n$, as required for consistency. We obtain the von Neumann entropy by analytic continuation and find agreement with the minimal surface prescription of Bousso and Penington.

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6 pages, 2 figures

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We propose that any compact ddd-manifold with elliptic data, J\mathcal{J}J, prepares a quantum state ∣J⟩|\mathcal{J}\rangle∣J⟩ on its (d−1)(d-1)(d−1)-boundary σσσ. Elliptic data consists of metric and field values, or their conjugates, but not both. No asymptotic structure is required. Inner products and traces are evaluated by the gravitational path integral with closed boundary conditions obtained by gluing elliptic data manifolds. In particular, we give a prescription for the Rényi entropies SnS_nSn​ of a subregion of σσσ. In a class of examples, we find that SnS_nSn​ is nonnegative and nonincreasing with nnn, as required for consistency. We obtain the von Neumann entropy by analytic continuation and find agreement with the minimal surface prescription of Bousso and Penington.
We associate a gravitational Hilbert space Hσ\mathbf{H}_σHσ​ to any closed compact (d−1)(d-1)(d−1)-manifold σσσ with real metric. A quantum state J(σ)\mathcal{J}(σ)J(σ) is a ddd-manifold bounded by σσσ and equipped with elliptic data. An inner product is defined by gluing states pairwise across σσσ and evaluated by viewing the resulting closed ddd-manifold as a boundary condition on the gravitational path integral (GPI) over (d+1)(d+1)(d+1)-manifolds. If σσσ is nonempty and the GPI is dominated by a single (d+1)(d+1)(d+1)-manifold MMM in the GN→0G_N\to 0GN​→0 limit, then MMM contains a Lorentzian CRT fixed-point set, providing J(σ)\mathcal{J}(σ)J(σ) with a classical spacetime interpretation. Conversely, given a finite Lorentzian domain with edge σσσ, a state J(σ)\mathcal{J}(σ)J(σ) may be associated to it by deforming its initial data off the real Lorentzian section and retaining only elliptic data. This establishes a broad correspondence between non-asymptotic spacetime regions and quantum states. Assuming that Hσ\mathbf{H}_σHσ​ factorizes over connected components of σσσ, our framework admits operators and partial traces. This allows us to explore the information-theoretic structure of the states we define. As an example, we construct a family of states by deforming partial Cauchy slices ΣΣΣ that straddle a two-sided black hole; σσσ consists of two spheres. We construct the reduced state on one sphere and find that its Rényi entropies are positive, monotonic, and sensitive to all aspects of ΣΣΣ and its complex deformation. The von Neumann entropy, however, is controlled only by the maximin surface in the causal domain of ΣΣΣ, independently of other parameters, so long as the complex deformation does not vanish. Our proposal may thus explain the efficacy of tensor network toy models of holography while transcending their limitations.
Entanglement entropy, or von Neumann 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 replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, 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. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and 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.
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What are gauge-invariant local observables in a subregion in quantum gravity? How does one even define such a subregion non-perturbatively? We study these questions in JT gravity. One can define a subregion by specifying the value of the dilaton at the boundary of the region. We study conformal matter correlators in such a subregion. There is a gravitational constraint associated with York time evolution within the causal diamond of the subregion. This constraint can be leveraged to construct gauge-invariant observables in quantum gravity, using a crossed product construction. The extrinsic curvature of Cauchy slices acts as the physical clock. This is a simple example of how gauge-invariant observables can be obtained by dressing to features of a spacetime (or other fields), without the need for introducing an external observer. The entropy associated with this algebra of observables is not an area, or any boundary term. We show that gravitational constraints only give boundary formulas for entropy when gauging isometric diffeomorphisms. York time flow is merely a conformal isometry, not an actual isometry, and thus leads to bulk contributions to entropy. We repeat our construction for Milne-type closed Big-Bang universes, which may be of independent interest.
In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy hatSn\\hat{S}_nhatSn​ that is related to the Renyi entropies SnS_nSn​, as given by the area of a \\emph{cosmic brane} minimally coupled with gravity, with a tension related to nnn that vanishes as nto1n\\to1nto1, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry. In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Renyi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Renyi entropies (SnS_nSn​) related to a minimal surface in the nnn replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.