Fixed-Area States in Quantum Gravity

Updated 9 November 2025

Fixed-area states are quantum states defined by sharply fixing the area of an extremal surface, central to studies in holographic quantum gravity.
They yield a flat entanglement spectrum where all Renyi entropies equal the fixed area divided by 4G, simplifying the analysis of gravitational path integrals.
Their construction aids in understanding entanglement phase transitions and holographic error correction through precise area projection techniques.

Fixed-area states are quantum states constructed by projecting a gravitational or field-theoretic system onto eigenspaces of a geometric operator—typically the area of a codimension-2 extremal surface such as the Ryu-Takayanagi (RT) or Hubeny-Rangamani-Takayanagi (HRT) surface in AdS/CFT. These states, which have suppressed or vanishing fluctuations in the specified area, serve as central tools for understanding entanglement entropy, modular structure, and semiclassical geometry in theories of holographic quantum gravity. The fixed-area construction has deep consequences for the structure of the entanglement spectrum, the efficiency of gravitational path integrals, holographic error correction, and the smoothing of phase transitions in entanglement entropy.

1. Formal Definition and State Construction

A fixed-area state $|\psi_{A_*}\rangle$ is a quantum state in which the area operator $\widehat{A}_{\gamma_R}$ associated with a codimension-2 extremal surface $\gamma_R$ (for example, an RT or HRT surface homologous to a boundary region $R$ ) is sharply fixed at $A_*$ . This is operationally achieved by inserting a projection operator or a delta-functional constraint into the gravitational or CFT path integral: $|\psi_{A_*}\rangle = \frac{P_{A_*} |\psi\rangle}{\sqrt{\langle\psi|P_{A_*}|\psi\rangle}}$ where $P_{A_*}$ projects onto the area eigenvalue $A_*$ . In Euclidean gravity, the path integral localizes on a geometry with a codimension-2 conical defect on $\gamma_R$ with the area set to $A_*$ . In the semiclassical (small $G$ ) regime, fluctuations in $A$ are suppressed ( $\Delta A \ll A_*$ ), so the RT entropy functional becomes exact, and the entanglement spectrum is modified in a universal way (Dong et al., 2018, Dong et al., 2022).

In the AdS/CFT context, fixed-area states in the CFT can be constructed by decomposing the reduced density matrix $\rho_A$ of a subregion $A$ in terms of its modular energy eigenbasis. The state is then projected onto a single modular energy $t$ value, yielding a state $|t\rangle_A$ whose dual geometry has a fixed area for the corresponding extremal surface (Guo, 2021).

2. Entanglement Spectrum and Flatness

At leading order in $1/G$, the entanglement spectrum of fixed-area states is flat: all Renyi entropies $S_n$ are independent of $n$ ,

$S_n(\rho_{R,\hat A}) = \frac{\hat A}{4G}$

This property arises because, after area projection, the replica trick path integral evaluates to a geometry with a fixed conical deficit, whose action contributions are linear in the area. Consequently, in gravitational path integrals, the reduced density matrix is (to leading order) proportional to the identity on the fixed-area subspace. The $n$ -dependence in Renyi entropies for generic (unprojected) states arises from integrating over fluctuating areas, with different $n$ favoring different saddle-points for the area (Dong et al., 2018, Guo, 2021).

Table: Entropic Features of Fixed-Area States

Property	Fixed-Area State	Generic State
Leading $S_n$ ( $n$ -indep)	$S_n = \hat{A}/4G$	$S_n = A_n / 4G$
Spectrum	Flat	Non-flat
Dominant Fluctuations	Area suppressed	Area fluctuations

This flatness underpins the approximation of entanglement spectra by random tensor network (RTN) models, where projective cuts (fixed-geometry) yield flat entanglement spectra per cut bond (Dong et al., 2018, Held et al., 4 Jan 2024).

3. Area Operators and Central Decompositions

In the algebraic formulation, the area operator $\widehat{A}$ lies in the center of the operator algebra associated to the entanglement wedge. The full code Hilbert space admits a central decomposition: $\mathcal{H} = \bigoplus_\alpha \mathcal{H}_{r_\alpha} \otimes \mathcal{H}_{\bar r_\alpha}$ where $\alpha$ labels area eigenvalues. A fixed-area state is obtained by projecting onto a single $\alpha$ block. In the context of exact quantum error-correcting codes (QECCs) without bulk local degrees of freedom, the central area operator vanishes. Upon coarse-graining—a binning of central eigenvalues—a nonzero, emergent area operator arises: $\widehat{A} = \int d\alpha\, \log n_\alpha \sum_O f_\alpha(O) |O\rangle\langle O|$ where $n_\alpha$ counts states in bin $\alpha$ , and $f_\alpha(O)$ is a binning function. Fixed-area states correspond to sharp projectors onto these coarse-grained labels. The resulting entanglement entropy reproduces RT/HRT in the semiclassical regime, modulo $\mathcal{O}(1)$ ambiguities from bin choices (Soni, 3 Nov 2025).

4. Path Integral Preparation, Diagonal Approximation, and Operator Commutativity

Fixed-area states are naturally prepared in the gravitational path integral using either Lagrange multipliers or delta constraints in the integral over bulk fields. The resolution of the identity in the Hilbert space can be written as an integral over fixed-area projectors. Crucially, when multiple areas are fixed, the “diagonal approximation” asserts that the resulting fixed-area sectors are orthogonal unless all areas match: $\langle \psi_{A_1, A_2, \dots}| \psi_{A'_1, A'_2, \dots} \rangle \approx \delta(A_1 - A'_1)\,\delta(A_2 - A'_2) \dots$ This holds when the corresponding area operators commute semiclassically. In AdS $_3$ gravity, explicit calculations show that in certain networks (e.g., simple 4-link “constrained HRT” configurations), all link-area operators commute, enabling simultaneous area projection (Held et al., 4 Jan 2024). However, in more general, nontrivial networks—such as those involving entanglement-wedge cross sections—the area operator algebra is non-Abelian, precluding sharply defined simultaneous fixed areas except in special cases.

5. Holographic Interpretation and Impact on Entanglement Entropy

Fixed-area states provide a clean semiclassical code subspace in holographic theories where the Ryu-Takayanagi formula holds exactly at leading order: $S(\rho_A) = \frac{\langle \psi | \widehat{A} | \psi \rangle}{4G}$ These states are fundamental to tensor-network toy models of AdS/CFT (where they mirror the fixed-geometry cuts) and to the understanding of modular flow (JLMS relations). The suppression of area fluctuations in fixed-area states means that, in the semiclassical limit, entanglement entropy simplifies to the area law.

Furthermore, decomposition into fixed-area sectors is key to understanding and smoothing RT/HRT “phase transitions.” Near an RT transition (where two saddle areas compete), area fluctuations in the path integral lead to an $O(G^{-1/2})$ correction in the entropy—a rounded crossover rather than a sharp kink, which is larger and of different order than bulk-quantum corrections ( $O(G^0)$ ). This smooths the entropy curve and matches predictions from random matrix and chaotic many-body models (Marolf et al., 2020).

6. Bulk Spacetime Geometry and Quantum Field Regularity

The spacetime geometry dual to a fixed-area state typically involves conical defects or generalized saddles with a codimension-2 area constraint. In Lorentzian signature, such geometries are real and free of conical singularities at nonzero time, but their curvature invariants can display power-law divergences along null congruences from the fixed-area surface. These divergences are “weak” at the classical level but become problematic for quantum fields, whose stress-energy can diverge as $U^{-2}$ near the surface.

To regulate such divergences, it is necessary to smear the area constraint over a transverse region of minimal width $\ell_{\rm sm} \sim \sqrt{G}$ , dictated by a semiclassical uncertainty relation $\Delta A \Delta s \gtrsim O(G)$ . Only such smeared fixed-area states are well-defined at the quantum level beyond leading $G$ order (Dong et al., 2022).

7. Extensions, Universality, and Connections to Lattice Models

Beyond the holographic setting, fixed-area states arise in statistical and lattice models (e.g., convex and column-convex polygons with fixed area), where analogous “entropic locking” phenomena occur. In the strongly inflated (small fugacity) regime, the mean perimeter approaches the minimal value (e.g., for a square in two dimensions), with universal asymptotic corrections reflecting the universality of area constraints (Mitra et al., 2010).

Fixed-area constructions also generalize to higher dimensions, more general quantum error correction models, and settings lacking bulk local modes, where emergent area corresponds to coarse-grained central elements in the boundary theory (Soni, 3 Nov 2025).

8. Ambiguities, Semiclassicality, and Open Problems

The definition of fixed-area states involves subtle ambiguities: choices of window functions or bin widths produce $\mathcal{O}(1)$ shifts in the definition of area eigenstates and resulting entropies. These ambiguities match analogous ones in the gravitational path integral. The validity of the diagonal approximation, the full classification of simultaneously commuting area operators, and the effect of quantum corrections beyond semiclassical leading order are active areas of research (Held et al., 4 Jan 2024, Soni, 3 Nov 2025). The semiclassical limit justifies many simplifications, but systematic corrections and precise operator algebras remain crucial for a complete understanding.

Fixed-area states thus constitute a rigorous, essential framework in gravitational path integrals, holographic entanglement, and quantum error correction, providing deep insights into the interface of geometry and information in quantum gravity.