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arXiv:2511.01382

Emergent Area Operators in the Boundary

Ronak M Soni

hep-thquant-phSubmitted Nov 3, 2025• Announced Nov 4, 2025• 0 citations• Revised Nov 4, 2025 (v2)
v1 · Nov 3, 2025v2 · Nov 4, 2025v3 · Aug 19, 2026
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Abstract

In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a central decomposition. However, the area operator that arises from this central decomposition vanishes. A non-zero area operator, however, emerges after coarse-graining. The expectation value of this operator approximates the actual entanglement entropy for a class of states that do not form a linear subspace. These non-linear constraints can be interpreted as semiclassicality conditions. The coarse-grained area operator is ambiguous, and this ambiguity can be matched with that in defining fixed-area states.

Comments
20 pages, comments welcome

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•Aug 12, 2026•0 citations
In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a central decomposition. However, the area operator that arises from this central decomposition vanishes. A non-zero area operator, however, emerges after coarse-graining. The expectation value of this operator approximates the actual entanglement entropy for a class of states that do not form a linear subspace. These non-linear constraints can be interpreted as semiclassicality conditions. The coarse-grained area operator is ambiguous, and this ambiguity can be matched with that in defining fixed-area states.
The fixed area states are previously discussed in the quantum error-correction codes interpretation of AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT. The dual bulk geometry is constructed by gravitational path integrals. In this paper we show the fixed area states correspondence in conformal field theories (CFTs), which are associated with the spectrum decomposition of reduced density matrix ${\ensuremath{\rho}}_{A}$ for a subsystem AAA. For two-dimensional CFTs we directly build the bulk metric, which is consistent with the expected geometry of the fixed area states. For arbitrary pure state $|\ensuremath{\psi}⟩$ with a geometric dual in the bulk we also find the consistency by using the gravity dual of R\'enyi entropy. We also obtain the parameters relation between the bulk geometry and boundary state. The pure state $|\ensuremath{\psi}⟩$ can be expanded as a superposition of the fixed area states. Motivated by this, we propose an area operator ${\stackrel{^}{A}}^{\ensuremath{\psi}}$. The fixed area state is the eigenstate of ${\stackrel{^}{A}}^{\ensuremath{\psi}}$, the associated eigenvalue is related to the R\'enyi entropy of subsystem AAA in this state. The Ryu-Takayanagi formula can be expressed as the expectation value $⟨\ensuremath{\psi}|{\stackrel{^}{A}}^{\ensuremath{\psi}}|\ensuremath{\psi}⟩$ divided by 4G4G4G, where GGG is the Newton constant. We further show the fluctuation of the area operator in the geometric state $|\ensuremath{\psi}⟩$ is suppressed in the semiclassical limit $G\ensuremath{\rightarrow}0$.
Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing subsystem erasure-correcting holographic code models describe only quantum fields on a fixed background: in such codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem complementary recovery and that incorporating backreaction naturally leads to approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the changes in extremal surface areas in gravitational systems. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.
Abstract Quantum extremal surfaces are central to the connection between quantum information theory and quantum gravity and they have played a prominent role in the recent progress on the information paradox. We initiate a program to systematically link these surfaces to the microscopic data of the dual conformal field theory, namely the scaling dimensions of local operators and their OPE coefficients. We consider CFT states obtained by acting on the vacuum with single-trace operators, which are dual to one-particle states of the bulk theory. Focusing on AdS3/CFT2, we compute the CFT entanglement entropy to second order in the large c expansion where quantum extremality becomes important and match it to the expectation value of the bulk area operator. We show that to this order, the Virasoro identity block contributes solely to the area operator.
One of the boldest ideas in modern physics is that space itself is a kind of error-correcting code. In this picture the interior of a region of spacetime is like a message, and the surface surrounding it is an encoded copy that stores the message redundantly, so that losing part of the surface need not lose the interior. A celebrated formula from holography then says something unexpected: the amount of entanglement carried by a boundary region equals the area of a surface reaching into the interior. This paper gives a complete, elementary proof that this area law is not an extra assumption bolted on by hand, but an unavoidable consequence of the code structure itself. The geometry is simply bookkeeping for redundancy. The derivation reduces a statement about gravity to plain facts about entanglement and about the algebra of operations one can perform on the encoded information. It establishes four things, each from scratch. First, in the simplest case it recovers the area law directly: the entanglement of a boundary region is a fixed geometric term plus the entanglement of whatever the interior itself carries. Second, it proves a symmetry that had previously been handled case by case, namely that a pure joint state assigns the same entanglement to a region and to its complement, and it proves this in full generality using a single clean identity about matrices. Third, it extends the area law to the realistic situation in which the code splits into separate sectors, each carrying its own independent area, so that the geometric term is not a single number but a genuine operator averaged over the sectors. Fourth, it proves the structural fact the whole picture rests on: the algebra of logical operations always decomposes cleanly into independent blocks, which is exactly the sector structure the area law needs. That last step comes down to one small analytic fact, that a matrix which is symmetric in the appropriate sense and also nilpotent must be zero, and the paper proves it directly. A companion result closes the reconstruction side of the story, showing that the operations recoverable from a boundary region are precisely those left untouched by the coarse-graining onto sectors, and that the coarse-grained state already reproduces every such measurement. Every step is elementary and self-contained, and the entire logical core has a companion version reconstructed inside a machine proof assistant, checked line by line with no gaps and resting only on the standard axioms of mathematics. The upshot is that a piece of geometry, an area, can be read off from nothing more than how redundantly a code stores its information. None of the underlying mathematics is new; it traces back to established results on holographic codes, to the entropy formula itself, and to the century-old classification of matrix algebras. What is new is the assembly: a short, honest, self-contained line of reasoning that turns a claim about spacetime into a handful of reusable facts about entropy and matrices, with every link exposed and checkable. Where the picture still reaches past what has been proved, in particular the finer tensor structure hidden inside each sector, the paper marks the boundary plainly instead of hiding it.