Entanglement-Wedge Reconstruction

Entanglement-wedge reconstruction says that a boundary region $A$ can represent a specified algebra of dressed bulk observables in its entanglement wedge $a=\operatorname{EW}(A)$, even when part of $a$ lies outside the causal wedge. The strongest controlled argument combines the generalized-entropy wedge with relative-entropy matching and quantum recovery on a semiclassical code subspace. It is not an exact finite-$N$ theorem about every bulk operator, nor does it identify one region-independent gravitational operator without fixing dressing, algebra, and error norm.

Required background. Entanglement Wedges, Nesting, and Information Inequalities defines $\operatorname{EW}(A)$ and its nesting properties. Causal Wedges and Subregion Reconstruction supplies the causal-wedge benchmark.

Helpful background. Operator-Algebra Quantum Error Correction gives the algebraic recovery criterion. Boundary Relative Entropy and Bulk Modular Data supplies the JLMS input. Quantum Communication and Entanglement Distribution and Reconstruction from Redundant Encodings clarify channel and redundancy language.

Reconstruction beyond the causal wedge

Let $V:\mathcal H_{\mathrm{code}}\to\mathcal H_{\mathrm{CFT}}$ be an encoding of a low-energy code subspace, $\mathcal M_a$ a von Neumann algebra of gauge-invariant observables associated with $a$, and $\mathcal N_A$ the boundary algebra in the domain of dependence of $A$. Exact algebraic reconstruction would mean that for every $O\in\mathcal M_a$ there is an $O_A\in\mathcal N_A$ such that

$$O_A V=VO, \qquad O_A^\dagger V=VO^\dagger .$$

The causal wedge $\operatorname{CW}(A)$ is determined by boundary-to-bulk causal propagation. The entanglement wedge is instead bounded by $A$ and the appropriate extremal or quantum extremal surface. A point can therefore satisfy

$$x\notin \operatorname{CW}(A), \qquad x\in \operatorname{EW}(A).$$

Ordinary causal HKLL support on $A$ then fails, while modular flow or recovery can still provide an $A$-supported representative. The original operator-algebra argument derives wedge reconstruction from equality of bulk and boundary relative entropies on the code subspace Dong, Harlow, and Wall 2016. Universal recovery channels make the information-theoretic implication quantitative Cotler et al. 2019.

From relative entropy to a recovery map

For code states $\rho$ and $\sigma$, suppose that the reduced boundary channel $\mathcal E_A$ and the bulk restriction to $a$ obey, to the stated semiclassical order,

$$D(\rho_A\lVert\sigma_A) =D(\rho_a\lVert\sigma_a)+O(\varepsilon).$$

Data processing gives $D(\rho_A\lVert\sigma_A)\leq D(\rho\lVert\sigma)$; near-saturation means that information discarded by restricting to $A$ is small for the chosen state family. A rotated Petz map built from a faithful reference state $\sigma$ has schematic form

$$\mathcal R_{\sigma,A}(X) =\int_{-\infty}^{\infty}\!\mathrm dt\,\beta(t)\, \sigma^{\frac{1-it}{2}} \mathcal E_A^\dagger\!\left( \sigma_A^{-\frac{1-it}{2}}X\sigma_A^{-\frac{1+it}{2}} \right) \sigma^{\frac{1+it}{2}},$$

with a normalized kernel $\beta(t)$. Recovery theorems bound a fidelity or relative-entropy loss, not an unqualified operator equality. Turning the recovered state channel into an operator representative also requires specifying the code, algebra, and topology or norm in which the adjoint channel is close.

First application

Take a static AdS geometry and a boundary interval $A$ whose extremal surface encloses a causal shadow. Choose a scalar wavepacket $\phi(f)$ supported near a point $x$ in that shadow and restrict the code to the vacuum plus finitely many low-energy wavepackets whose backreaction is $G_N E/L^{d-2}\ll1$. Causal smearing over $D[A]$ cannot reach $x$, but the relative-entropy relation for perturbations generated by $\phi(f)$ licenses recovery from $A$ at leading semiclassical order.

A reproducible statement records the dressed operator, the reference state, the energy cutoff, and the recovery error. For example,

$$\sup_{\rho\in\mathcal S_{E_*}} \left\lVert \mathcal R_A\mathcal E_A(\rho)-\rho \right\rVert_1\leq\varepsilon_A(E_*,N),$$

rather than saying merely that “the point is encoded in $A$.” The comparison establishes a wedge-sized recovery domain in the semiclassical state family; it does not turn $\phi(f)$ into an asymptotic S-matrix observable.

Adversarial control

Choose two boundary regions whose semiclassical wedges both contain the support of $\phi(f)$ and demand one exactly identical, code-preserving finite-$N$ logical operator with representatives in both regions. The demand forces that logical operator to commute, on the code, with both complementary local algebras. A finite-dimensional toy code can have such an invisible protected sector; a gravitational CFT additionally has dressing, stress-tensor detection, and code leakage. The 2026 preprint by Terashima argues that these effects obstruct a nontrivial shared finite-$N$ logical algebra and favor region-adapted reconstruction Terashima 2026. That is a material counterclaim, not a settled theorem about all holographic theories.

The control therefore downgrades the conclusion to region- and code-dependent approximate reconstruction unless a common algebra, dressing prescription, and error bound are independently established.

Regime, evidence ceiling, and handoff

The usual controlled hierarchy is $G_N/L^{d-1}\sim N^{-2}\ll1$, low code energy, fixed operator number, a bulk EFT cutoff below the string and Kaluza–Klein scales, and surfaces separated far enough from a phase transition that $1/N$ corrections do not exchange the dominant wedge. String corrections scale with powers of $\alpha'/L^2$; bulk loops scale with $G_N$; errors can become nonuniform when the code dimension grows exponentially with $N^2$ or when competing generalized entropies differ by $O(1)$.

The evidence ceiling is leading semiclassical relative-entropy and recovery control for a specified algebra. Exact toy-code recovery, approximate finite-$N$ recovery, and a shared gravitational logical algebra are separate claims. Continue to Leading Semiclassical JLMS and Code-Subspace Claims for the modular relation and to Approximate Finite-N Recovery, Alpha-Bits, and Error Bounds for explicit error metrics.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Cotler, J., Hayden, P., Penington, G., Salton, G., Swingle, B., and Walter, M. (2019), “Entanglement Wedge Reconstruction via Universal Recovery Channels,” Physical Review X 9, 031011. DOI; arXiv:1704.05839.
• Dong, X., Harlow, D., and Wall, A. C. (2016), “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Physical Review Letters 117, 021601. DOI; arXiv:1601.05416.
• Jafferis, D. L., Lewkowycz, A., Maldacena, J., and Suh, S. J. (2016), “Relative Entropy Equals Bulk Relative Entropy,” Journal of High Energy Physics 2016(06), 004. DOI; arXiv:1512.06431.
• Terashima, S. (2026), “Entanglement Wedge Reconstruction without Holographic Quantum Error Correction,” preprint. arXiv:2607.08684.