Entanglement-Wedge and Interior Reconstruction

Bulk reconstruction asks when an operator in a semiclassical region can be represented by an operator in a boundary, exterior, or radiation algebra. The answer is never just “inside the entanglement wedge.” One must declare the code subspace, encoding, logical algebra and center, physical region, state family, norm, and accuracy. This chapter builds the exact finite-dimensional recovery statements first, connects them to generalized-entropy wedges, and only then assesses mirror operators, island reconstruction, non-isometric models, and nonperturbative failure.

Helpful background. Semiclassical Gravity supplies QES wedges and generalized entropy; Quantum Information supplies channels, recovery, relative entropy, and operator algebras; Quantum Fields in Curved Spacetime supplies effective bulk observables; Black-Hole Information supplies islands and factorization questions.

Operators on a code subspace

Let $\mathcal H_{\mathrm{code}}$ be a regulated set of semiclassical states and let

$$V:\mathcal H_{\mathrm{code}} \longrightarrow \mathcal H_A\otimes\mathcal H_{\bar A}$$

be an isometric encoding for the elementary theorem setting. A logical operator $O$ is reconstructible on $A$ if there exists $O_A$ such that

$$O_A V=VO$$

on the declared code subspace. In an approximate setting, equality is replaced by a named norm bound, for example

$$\sup_{\substack{\lvert\psi\rangle\in\mathcal H_{\mathrm{code}}\\ \lVert\psi\rVert=1}} \left\lVert(O_AV-VO)\lvert\psi\rangle\right\rVert \le\varepsilon.$$

The supremum is over normalized states: without that restriction, rescaling a vector makes any nonzero error unbounded. The state family and norm are part of the claim. An operator that works on one equilibrium state but fails on superpositions is not a state-independent reconstruction on their span.

Gravity often requires an operator algebra $\mathcal M$ rather than a tensor factor. Its center can contain an area operator or charge data. Complementary recovery then assigns $\mathcal M$ to $A$ and its commutant to $\bar A$ without claiming that central information belongs exclusively to either side. The code-subspace and multiple-representation viewpoint was introduced holographically in Almheiri, Dong, and Harlow 2015, §§2–4.

Routes through the chapter

The reading sequence is Code Subspaces and Approximate Bulk Operators, Quantum Error Correction and Complementary Recovery, Entanglement-Wedge Reconstruction, Interior and Mirror Operators, Island Reconstruction from Radiation, and Reconstruction Accuracy and Nonperturbative Failure.

The theorem route takes the first three pages. It proves recovery from complementary-channel decoupling, then uses equality of bulk and boundary relative entropy to obtain entanglement-wedge reconstruction. This route is finite-dimensional at the proof step and adds gravitational $1/N$ qualifications afterwards.

The interior route begins with code subspaces, then reads mirror operators, island reconstruction, and the accuracy page. It compares thermofield-double continuation, modular constructions, state dependence, complexity protection, and the radiation algebra. It is deliberately more cautious because several maps are effective, approximate, or state-family dependent.

Relative entropy and wedge recovery

For code states $\rho$ and $\sigma$, JLMS relates boundary and bulk relative entropy for a boundary region $A$ and its wedge $a$,

$$S(\rho_A\Vert\sigma_A) =S(\rho_a\Vert\sigma_a)+O(1/N).$$

The area or center contribution cancels in the relative-entropy difference under the stated code assumptions. Jafferis, Lewkowycz, Maldacena, and Suh 2016, §§2–4 derive the modular and relative-entropy relation. In a finite-dimensional regulated setting, equality for all code states implies a recovery representation of the wedge algebra; the proof is given in Dong, Harlow, and Wall 2016, complete Letter and proof.

The theorem does not make recovery computationally easy. It also does not extend one operator representation to arbitrary high-energy states. Enlarging the code changes the complementary information and can destroy correctability.

Frontier extensions

Non-isometric code models distinguish fundamental unitary black-hole dynamics from an effective interior map that is not an isometry on its whole nominal domain. In random models, the map can approximately preserve inner products and expectation values on subexponential state families while failing for exponential families or sufficiently complex operations. The construction, QES relation, and complexity protection are developed in Akers, Engelhardt, Harlow, Penington, and Vardhan 2022, §§2–8. Its random-model, state-family, norm, complexity, and code-domain assumptions must accompany any conclusion.

A newer position argues that islands and reconstruction need not rely on an auxiliary nongravitating bath. Antonini, Chen, Maxfield, and Penington 2025, §§2–4 analyze massless-gravity and asymptotically flat examples using specified asymptotic algebras, background symmetries, many-copy measurement protocols, and gravitational observables that are compact to all perturbative orders. The radiation representative can be a microscopic, nonperturbative operator, but its agreement with the island operator is controlled only to the perturbative order at which that semiclassical operator is defined. Exact nonperturbative bulk compact support, beyond-all-orders error control, operational cost, approximate symmetries, and the precise algebra remain qualifications.

Results and open questions as of 2026-08-19

This discussion reflects results available through 2026-08-19.

Evidence or result	Controlled scope	Limitation or unresolved inference
Holographic code subspaces, Almheiri, Dong, and Harlow 2015, §§2–4	Finite semiclassical code subspace and erasure-correction analogy	Tensor-network illustrations do not prove recovery on the full Hilbert space
Relative-entropy reconstruction, Dong, Harlow, and Wall 2016, complete Letter and proof	Regulated code with relative-entropy equality for the relevant algebras	Approximate gravitational equality requires an explicit error conversion
Non-isometric interior codes, Akers et al. 2022, §§2–8	Random effective map, subexponential state family, bounded complexity, declared norm	Fails for outside-code or exponentially complex operations; fundamental dynamics remains unitary
Islands without auxiliary baths, Antonini et al. 2025, §§2–4	Massless gravity, specified asymptotic algebras, no-isometry or controlled-symmetry backgrounds, many-copy access, perturbatively compact bulk operators, and microscopic radiation representatives	Exact nonperturbative bulk compact support, beyond-all-orders matching, and resource-feasible measurement remain open

A conservative position restricts reconstruction to a small code subspace and controlled operator algebra. A permissive position treats QES wedges and complexity protection as evidence for a much larger effective interior. State-dependent proposals restrict which state family shares an operator assignment; non-isometric proposals can retain a linear map while giving up exact inner-product preservation on its whole effective domain. These are different modifications. Their domain and accuracy must be stated separately; no universal full-Hilbert-space reconstruction is established.

An adverse check enlarges the state set until the complementary subsystem can distinguish logical data. Exact recovery then fails by the information–disturbance theorem. In non-isometric models, exponentially many states or highly complex operations similarly expose inner-product distortion. In asymptotic-gravity proposals, changing the background symmetry or allowed algebra can remove perturbative compactness.

The conclusions could change after a theorem converts gravitational relative-entropy errors into sharp operator-norm recovery bounds, after a no-auxiliary-bath protocol survives nonperturbative and resource checks, or after a controlled model shows that exponential state families or outside-code operations remain reconstructible.

Review the chapter

Diagnostic. Encode $\lvert0\rangle\mapsto\lvert00\rangle$ and $\lvert1\rangle\mapsto\lvert11\rangle$. Which logical Pauli operator can either one-qubit region represent, and which cannot? What does this teach about algebraic recovery?

Solution

Logical $Z$ is represented by physical $Z$ on either qubit because both act with eigenvalues $+1$ and $-1$ on the two codewords. No operator on one qubit represents logical $X$ on the full code: flipping one physical qubit sends a codeword outside the code, while a diagonal or general one-qubit action cannot interchange both global codewords without acting on the other qubit. The recoverable algebra can therefore be the commuting algebra generated by $Z$ even when the full logical matrix algebra is not recoverable.

References

• Akers et al. 2022, Chris Akers, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan, “The Black Hole Interior from Non-Isometric Codes and Complexity,” arXiv:2207.06536 (2022).
• Almheiri, Dong, and Harlow 2015, Ahmed Almheiri, Xi Dong, and Daniel Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” Journal of High Energy Physics 04 (2015) 163.
• Antonini et al. 2025, Stefano Antonini, Chang-Han Chen, Henry Maxfield, and Geoff Penington, “An Apologia for Islands,” arXiv:2506.04311 (2025).
• Dong, Harlow, and Wall 2016, Xi Dong, Daniel Harlow, and Aron C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Physical Review Letters 117 (2016) 021601.
• Jafferis et al. 2016, Daniel L. Jafferis, Aitor Lewkowycz, Juan Maldacena, and S. Josephine Suh, “Relative Entropy Equals Bulk Relative Entropy,” Journal of High Energy Physics 06 (2016) 004.