Reconstruction Accuracy and Nonperturbative Failure

A reconstruction map is accurate only relative to a declared state family, operator algebra, physical region, norm, and resource class. Exact recovery can hold on a finite code, while the same formula fails after the code is enlarged; perturbative agreement can coexist with nonperturbative error; and an operator may exist mathematically but require an infeasible decoder. The useful answer is therefore an error budget and a domain statement, not a universal claim that “the interior is encoded.”

Required background. Quantum Information supplies approximate channels, operator norms, and code-subspace recovery; Gauge Theory and String-Theory Preparation supplies holographic encoding, gauge constraints, and gravitational dressing.

Helpful background. Semiclassical Gravity supplies generalized entropy and quantum extremal surfaces; Quantum Fields in Curved Spacetime supplies the effective interior algebra; Black-Hole Information supplies Page transitions and the unitarity problem.

Four independent accuracy axes

Let $V$ encode a regulated code subspace $\mathcal C$ into the physical Hilbert space, and let $O$ be a bounded logical operator. For a proposed representative $O_A$ on region or algebra $A$, one useful worst-case error is

$$\varepsilon_O(\mathcal C) = \sup_{\substack{\lvert\psi\rangle\in\mathcal C\\ \lVert\psi\rVert=1}} \left\lVert \left(O_AV-VO\right)\lvert\psi\rangle \right\rVert .$$

This definition fixes the norm and requires one map to work on every superposition in $\mathcal C$. It is stronger than matching a few correlators or succeeding on a typical state.

Its operational meaning can be stated for a Hermitian bounded observable: an intertwining error at most $\varepsilon_O$ bounds the error in every normalized code-state expectation value by $\varepsilon_O$, by Cauchy–Schwarz. Recovering an entire state is a stronger claim and needs a channel-distance bound that also controls entanglement with a reference. The numerical value of an operator error also scales if $O$ is rescaled, so comparisons should fix a normalization such as $\lVert O\rVert_{\mathrm{op}}\le1$.

At least four contributions should be kept distinct. A semiclassical error comes from omitted orders in $G$, $1/N$, or the bulk loop expansion. A nonperturbative error, often parametrically of order $e^{-cS}$, is invisible to every fixed order but can matter after exponentially many trials or on exponentially large state families. A domain error appears when $O_A$ is applied outside the code or after backreaction changes the relevant QES. A resource error records that constructing or measuring $O_A$ may require exponential circuit size, precision, copies, or time. These terms need not add incoherently, so a controlled analysis bounds the chosen total norm rather than merely listing scalings.

Gravitational dressing changes the algebra

A compactly supported field $\phi(x)$ at a coordinate point is not a diffeomorphism-invariant gravitational observable. One may attach a line-like or Coulomb-like dressing to an asymptotic anchor, producing a relational observable $\Phi_\Gamma$, but the dressing carries a long-range field. In perturbative gravity about flat spacetime, operators with nonzero Poincaré charges require such dressing and the resulting physical observables need not form exactly commuting local subalgebras; the construction and commutator obstruction are given in Donnelly and Giddings 2016, §§2–5.

This result rules out silently importing an exact nongravitational tensor factor into gravity. It does not rule out approximate locality after a gauge, asymptotic anchor, charge sector, perturbative order, and dressing prescription have been fixed. Two dressings of the same effective field can differ by radiative gravitational data. Consequently, a reconstruction error for the dressed algebra is not automatically an error for an imagined bare local operator, and a leading-order commutator is not an exact nonperturbative factorization theorem.

Code size competes with uniform recovery

Suppose a boundary region $A$ has two candidate extremal surfaces of areas $A_1$ and $A_2\gt A_1$, and a black hole has entropy $S=A_0/(4G)$. For a microstate subspace of dimension $d$, mixing with a reference can contribute $\log d$ to the bulk entropy and move the minimizing surface. In the model analyzed by Hayden and Penington, the crossover occurs when

$$\log d+\frac{A_1}{4G} \gt \frac{A_2}{4G}, \qquad d\gtrsim e^{\alpha S}, \qquad \alpha=\frac{A_2-A_1}{A_0}.$$

Below this scale a single reconstruction can work on large subspaces; above it, the same region need not contain the operator in the entanglement wedge for all states entangled with a reference. The derivation and the requirement to test mixed states are in Hayden and Penington 2019, §1, Eq. (1.1), and §§4.1–4.2. Their §8.1 gives nonperturbative accuracy bounds and shows why error and code dimension cannot be varied independently.

The scaling is not a universal numerical threshold: $\alpha$ depends on the region and candidate surfaces, and the argument assumes the semiclassical QES prescription and the chosen microstate family. Its durable lesson is logical. Pointwise reconstruction for every small subspace does not imply one state-independent operator on the full black-hole Hilbert space.

Effective non-isometric interiors

Late-time effective field theory may assign an interior Hilbert space larger than the fundamental black-hole state space. A linear effective map

$$V_{\mathrm{eff}}: \mathcal H_{\mathrm{EFT}} \longrightarrow \mathcal H_{\mathrm{fund}}$$

then cannot be an isometry on its whole nominal domain. This does not make the fundamental time evolution nonunitary. In the non-isometric-code model of Akers, Engelhardt, Harlow, Penington, and Vardhan, the underlying dynamics is unitary, while projection or postselection produces an effective map that approximately preserves overlaps for selected subexponential state families. The random-map construction and concentration estimates are in Akers et al. 2022, §§2–3; state-specific, wedge, and subspace-dependent reconstruction are separated in their §§5.2–5.4; fundamental and effective dynamics are distinguished in §7.

Complexity protection is an operational statement, not exact norm equality. In that model, simple observers may be unable to resolve the null directions or overlap distortion that an exponentially complex operation exposes; see Akers et al. 2022, §§8.1–8.3. The conclusion depends on the random ensemble, chosen state family, circuit model, allowed measurements, and norm. It fails for unrestricted exponential families, operations outside the protected class, or states beyond the effective domain.

Results and open questions as of 2026-08-19

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

Evidence or result	Assumptions	Controlled result	Limitation
Relative-entropy reconstruction, Dong, Harlow, and Wall 2016, complete Letter and proof	Finite-dimensional regulator, isometric code, specified logical algebra and complementary regions, relative-entropy equality for all code states	An operator representative exists on the boundary region throughout the declared code	Does not itself convert finite-$N$ gravitational corrections into a sharp worst-case operator-norm error
Alpha-bit bounds, Hayden and Penington 2019, §1, Eq. (1.1), §§4.1–4.2, and §8.1	AdS/CFT microstate subspaces, QES prescription, reference systems, approximate correction	Relates recoverable subspace dimension to region geometry and gives nonperturbative accuracy constraints	Region-, geometry-, and code-family dependent; not a decoder-complexity bound
Dressing obstruction, Donnelly and Giddings 2016, §§2–5	Perturbative gravity about flat spacetime, asymptotic anchoring, declared gauge and dressing	Charged compact observables require long-range gravitational dressing, obstructing exact local commuting factors	Does not supply a general nonperturbative algebra for AdS interiors or evaporating black holes
Non-isometric code model, Akers et al. 2022, §§2–3 and §§5–8	Solvable random model, selected subexponential states, stated overlap norm and circuit-complexity class	Effective non-isometric encoding can reproduce QES, reconstruction, measurement, and complexity-protected predictions while fundamental dynamics stays unitary	Exponential or outside-code operations expose failures; the model is not a universal microscopic completion

The conservative position accepts exact finite-code theorems and perturbative wedge reconstruction but demands an explicit norm bound before extending them. The large-code position uses alpha-bit scaling to allow exponentially large yet still restricted microstate subspaces, with unavoidable nonperturbative error. The complexity-protected position permits a still larger effective interior because only feasible operations must agree. The global-isometry position demands one linear isometry and operator algebra on the full Hilbert space; simple dimension counting makes that demand incompatible with an oversized effective interior. These positions answer different domain and operational questions. No controlled result in the table establishes one exact, efficient reconstruction for all black-hole microstates and all operations; that extrapolation remains unresolved.

An adverse finite-dimensional check

Let $W:\mathbb C^3\to\mathbb C^2$ act on orthonormal input states by

$$W\lvert0\rangle=\lvert0\rangle, \qquad W\lvert1\rangle=\lvert1\rangle, \qquad W\lvert2\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

On $\mathcal C_2=\operatorname{span}\{\lvert0\rangle,\lvert1\rangle\}$, $W$ is exactly isometric. Enlarge the code by the named state $\lvert2\rangle$, and

$$\langle0\vert2\rangle=0, \qquad \langle W0\vert W2\rangle=\frac1{\sqrt2}.$$

Indeed $W\bigl(\lvert2\rangle-(\lvert0\rangle+\lvert1\rangle)/\sqrt2\bigr)=0$, so the enlarged map has a null direction and no inverse on $\mathbb C^3$. This changes only the code-subspace assumption and disproves a global reconstruction, while leaving the exact two-state result intact. Random high-dimensional models can hide analogous directions from low-complexity tests, but complexity does not remove the rank obstruction.

The conclusions could change after a theorem gives sharp gravitational operator-norm errors, a single-theory model controls exponential state families, an adverse example defeats complexity protection, or a decoding protocol survives resource and backreaction tests.

Exercises

A proposed representative obeys

$$\left\lVert \left(O_AV-VO\right)\lvert i\rangle \right\rVert\le\delta$$

for each vector in an orthonormal basis $\{\lvert i\rangle\}_{i=1}^{K}$. Is it justified to advertise uniform error $\delta$ on their span? Give the bound supplied by this information alone and reformulate the claim.

Solution

No. For $\lvert\psi\rangle=\sum_i c_i\lvert i\rangle$, the triangle inequality and Cauchy–Schwarz give

$$\left\lVert \left(O_AV-VO\right)\lvert\psi\rangle \right\rVert \le \delta\sum_i\lvert c_i\rvert \le \sqrt K\,\delta$$

for normalized $\lvert\psi\rangle$. Coherent errors can attain the larger scaling, so basis-state tests alone do not establish worst-case error $\delta$. The defensible statement is either “error at most $\delta$ on each tested basis state” or “uniform error at most $\sqrt K\,\delta$ on their span.” A stronger claim requires a direct operator-norm or channel-distance estimate.

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).
• 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.
• Donnelly and Giddings 2016, William Donnelly and Steven B. Giddings, “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems,” Physical Review D 94 (2016) 104038.
• Hayden and Penington 2019, Patrick Hayden and Geoffrey Penington, “Learning the Alpha-Bits of Black Holes,” Journal of High Energy Physics 12 (2019) 007.