Bulk Reconstruction Beyond Semiclassical Code Subspaces

The question is: How far can bulk observables and geometry be reconstructed at finite $N$, finite coupling, and outside a fixed semiclassical code subspace? Exact subregion reconstruction is well understood as an operator-algebra quantum-error-correction statement inside a chosen code subspace. At finite $N$ and coupling it becomes approximate and state-dependent through its error budget. No general construction recovers a single state-independent algebra of sharply local bulk operators on the entire boundary Hilbert space.

Evidence cutoff. 11 August 2026.

Required background. Finite-$N$ horizons, state dependence, and limits supplies the obstruction from gravitational dressing and finite entropy. Code subspaces, logical algebras, and encoding maps supplies the precise reconstruction problem.

Helpful background. Approximate finite-$N$ recovery supplies quantitative error notions. Operator-algebra quantum error correction explains why a subregion reconstructs an algebra rather than a tensor factor. Approximate recovery and information disturbance relates reconstruction error to leaked information. Large-gap locality tests supplies the spectral conditions for a local bulk effective theory.

Reconstruction as an encoding statement

Let $V:\mathcal H_{\rm code}\to\mathcal H_A\otimes\mathcal H_{\bar A}$ encode semiclassical states into the boundary. A logical operator $O$ is exactly reconstructable on $A$ when some $O_A$ obeys

$$O_A V\lvert\psi\rangle=VO\lvert\psi\rangle \quad\text{for every }\lvert\psi\rangle\in\mathcal H_{\rm code}.$$

For an algebra $\mathcal M$ associated with the entanglement wedge of $A$, the complementary-recovery condition says that information about $\mathcal M$ is available in $A$ while its commutant is available in $\bar A$. At finite $N$, an operational statement replaces equality by a channel distance or worst-case expectation-value error. The result is meaningful only after the code, algebra, norm, boundary region, and energy/time range are fixed.

Scope coordinate	Required declaration
Boundary theory	A specified holographic theory, $N$, coupling, state, and boundary region
Code	Energy band or family of semiclassical geometries and excitations
Bulk object	Dressed field, relational observable, logical algebra, or geometric functional
Accuracy	Diamond norm, entanglement fidelity, relative entropy, or a stated correlator norm
Time	Fixed-time wedge, causal evolution interval, or behind-horizon domain
Non-question	Whether a gauge-invariant local bulk tensor factor exists independently of dressing and state

Established results and their hypotheses

Perturbative reconstruction. In large-$N$ CFT states with a weakly coupled local bulk dual, HKLL-type smearing reconstructs free bulk fields and admits perturbative interaction corrections Hamilton, Kabat, Lifschytz, and Lowe 2006. The expansion presupposes a semiclassical background, a controllable spectrum, and an operator dictionary.

Code-subspace structure. The apparent ability to reconstruct the same bulk operator from different boundary regions is consistent because low-energy bulk states form a quantum code Almheiri, Dong, and Harlow 2015. Equality of bulk and boundary relative entropies implies entanglement-wedge reconstruction for an appropriate logical algebra Dong, Harlow, and Wall 2016.

Finite-error interpretation. Universal recovery maps turn approximate relative-entropy equality into an approximate decoder Cotler et al. 2019. These are theorem-level information-theoretic implications once their channel and state-set hypotheses hold; establishing the small error from a particular finite-$N$ CFT is a separate dynamical problem.

Interpretation. “The bulk is encoded on the boundary” means that selected logical observables have equivalent actions on a selected state set. It does not mean that every semiclassical chart, topology, and interior operator is simultaneously represented by one linear local algebra on the full Hilbert space.

Competing extensions beyond the code subspace

Position	Supporting evidence	Obstruction or qualification
Enlarge the code until it covers all semiclassical geometries.	Quantum error correction naturally supports direct sums of sectors, and operator-algebra codes accommodate centers and area operators. State-specific reconstruction can also be related to quantum minimal surfaces Akers and Penington 2022.	Entanglement wedges change discontinuously across QES transitions; a single reconstruction may fail when the logical algebra or geometry changes.
Finite-$N$ reconstruction is just perturbative HKLL with more terms.	$1/N$ and inverse-gap corrections systematically improve correlators in a fixed background.	The series need not be uniform in energy, time, or code dimension. Finite entropy, black-hole recurrences, and nonperturbative $e^{-N}$ effects are invisible at every finite perturbative order.
Approximate QEC supplies a complete nonperturbative definition.	It gives regulator-independent operational error criteria and does not require a particular tensor-network model.	It diagnoses an encoding once the channel and code are known; it does not derive them from the CFT or establish bulk locality.
State-dependent interior operators evade finite-entropy obstructions.	Restricting the state set can preserve smooth-interior correlators.	Nonlinearity, overlap consistency, and compatibility among different state patches must be demonstrated; otherwise predictions for superpositions are ambiguous.

No-go pressures and method limits

Gauge invariance forces gravitationally dressed operators to reach an asymptotic reference, so exact commutativity of sharply localized bulk algebras is not expected at finite gravitational coupling. Boundary completeness also prevents an exactly independent interior tensor factor. Quantum no-cloning forbids two boundary regions from each carrying an unrestricted exact copy; code-subspace restrictions and complementary logical algebras are what remove the contradiction.

Large $N$ is not sufficient for local bulk physics. A sparse spectrum/large higher-spin gap and suitable Mellin-amplitude behavior are additional locality tests. Tensor networks exhibit the encoding geometry vividly but do not determine finite-coupling CFT errors unless derived from the theory. Correlator matching can miss worst-case errors on a code whose dimension grows too quickly.

Status — controlled in fixed semiclassical codes, open globally. Exact and approximate reconstruction theorems give a precise framework, and perturbative bulk operators are calculable in favorable holographic theories. Uniform finite-$N$, finite-coupling reconstruction across changing geometries and beyond a bounded code subspace is not established.

Resolution criteria

A substantive extension would specify a concrete boundary theory and provide:

1. an explicit family of codes whose dimension and energy range scale with $N$;
2. relationally dressed bulk algebras and encoding/recovery channels defined without selecting the answer geometry by hand;
3. nonperturbative error bounds uniform over the full claimed state set and time interval;
4. compatibility across overlapping codes, QES transitions, topology changes, and superpositions; and
5. independent checks from spectra, OPE/Mellin data, correlators, relative entropy, and causal response.

Research connections

The field context is holography and quantum gravity. Holographic reconstruction and gravitational path integrals supplies the dictionary and saddle expansion, while replica, modular, and operator-algebra methods supplies the recovery criteria. Concrete benchmarks should verify perturbative HKLL smearing in a specified background, quantify recovery error under boundary erasures, and test whether large-$N$ factorization and a parametrically sparse low-dimension spectrum suffice for local bulk effective field theory in the chosen code subspace.

Literature selection and cutoff

The finite source set was selected through targeted journal, arXiv, and citation searches for constructive smearing, operator-algebra QEC, entanglement-wedge theorems, universal approximate recovery, and explicit limits outside fixed-area/code sectors available through 11 August 2026. Tensor-network illustrations were not treated as evidence about an unspecified CFT’s finite-$N$ error. The selection is not exhaustive.

References

• Akers, C., and Penington, G. (2022). “Quantum Minimal Surfaces from Quantum Error Correction.” SciPost Physics 12, 157. DOI; arXiv:2109.14618.
• Almheiri, A., Dong, X., and Harlow, D. (2015). “Bulk Locality and Quantum Error Correction in AdS/CFT.” Journal of High Energy Physics 2015, 163. DOI; arXiv:1411.7041.
• 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.
• Hamilton, A., Kabat, D., Lifschytz, G., and Lowe, D. A. (2006). “Local Bulk Operators in AdS/CFT: A Boundary View of Horizons and Locality.” Physical Review D 74, 066009. DOI; arXiv:hep-th/0506118.