Approximate Finite-N Recovery, Alpha-Bits, and Error Bounds

Finite-$N$ holographic recovery must be stated as an approximation in a named operational metric, on a bounded state family, for a specified boundary channel. Alpha-bit claims add a code-size scaling: one asks for recovery uniformly on every subspace whose dimension grows like $e^{\alpha S}$, not for one universal decoder on the entire black-hole Hilbert space. Perturbative equality to all orders in $1/N$ does not eliminate nonperturbative error, and current arguments disagree about whether a nontrivial region-independent logical algebra survives at ordinary finite $N$.

Required background. Approximate Recovery and Information–Disturbance supplies the error bounds. Finite N, Horizons, State Dependence, and Reconstruction Limits supplies the gravitational ceiling.

Helpful background. Infinite-Dimensional and Energy-Constrained Channel Distances, Energy-Constrained Capacities and Coding Tasks, and Energy-Constrained Errors and Infinite-Dimensional Norms provide continuum-safe metrics. Holographic Quantum Error Correction supplies the geometric noise model.

Recovery error and complementary leakage

For a code channel $\mathcal E_A$ and target logical restriction $\mathcal T_a$, define an energy-constrained recovery error

$$\varepsilon_{\mathrm{rec}}(E_*) =\frac12\inf_{\mathcal R_A} \left\lVert \mathcal R_A\circ\mathcal E_A-\mathcal T_a \right\rVert_{\diamond,E_*}.$$

The constrained diamond norm includes arbitrary reference-system entanglement but restricts inputs by $\operatorname{Tr}(H\rho)\leq E_*$. A weaker state-by-state trace distance cannot license a uniform operator-algebra statement.

Let $\mathcal E_{\bar A}$ be the complementary channel and $\mathcal C$ the ideal channel that retains only allowed center data. Information–disturbance theorems relate recovery and leakage schematically by

$$\frac14\left\lVert\mathcal E_{\bar A}-\mathcal C\right\rVert_{\diamond,E_*}^{2} \lesssim\varepsilon_{\mathrm{rec}}(E_*) \lesssim \sqrt{\left\lVert\mathcal E_{\bar A}-\mathcal C\right\rVert_{\diamond,E_*}}.$$

The constants depend on the precise metric and algebraic formulation. The important point is operational: reconstruction is good exactly to the extent that the erased region fails to distinguish the protected logical data.

Alpha-bits and code-size dependence

Let a black-hole sector have entropy $S_{\mathrm{BH}}=O(N^2)$. An $\alpha$-bit task asks for accurate transmission or reconstruction on arbitrary subspaces $\mathcal H_s$ satisfying

$$\log\dim\mathcal H_s\leq\alpha S_{\mathrm{BH}}, \qquad 0\leq\alpha\leq1,$$

with a decoder allowed to depend on the chosen subspace. This interpolates between recovery of fixed small subspaces and full quantum information. Hayden and Penington showed that holographic black-hole channels exhibit such regimes and derived nonperturbative lower bounds on reconstruction error Hayden and Penington 2019.

Alpha-bit capacity is therefore not the statement that a fraction $\alpha$ of one fixed logical algebra is exactly shared. It is a family of subspace transmission tasks with specified error and decoder dependence.

First application

Fix a microcanonical band with entropy $S$ and select subspaces of dimension $d_s=e^{\alpha S}$. For each $\mathcal H_s$, evaluate the complementary leakage

$$\delta_s =\sup_{\rho,\sigma\in\mathcal S(\mathcal H_s)} \frac12\left\lVert \mathcal E_{\bar A}(\rho)-\mathcal E_{\bar A}(\sigma) \right\rVert_1.$$

Use the same energy band, region, center convention, and gravitational dressing for every comparison. Report the largest $\alpha$ for which a uniform recovery target $\varepsilon_0$ is met, together with whether the decoder is universal or subspace dependent. This is an alpha-bit result; it is not full finite-$N$ entanglement-wedge reconstruction.

Adversarial control

First increase $\alpha$ until $d_s$ is exponentially large enough that small pairwise errors accumulate; a fixed-order $1/N$ estimate can then cease to be uniform. Next ask for one shared, code-preserving logical operator on multiple boundary regions, with one gravitational dressing and a finite-$N$ norm bound. The 2026 preprint by Terashima argues that local stress-tensor observables obstruct such a protected common algebra in an ordinary finite-$N$ CFT Terashima 2026. By contrast, approximate-code analyses show controlled recovery and state-dependent area behavior in other precisely defined models Cao et al. 2026, and Witten obtains a scale separation between area-function and recovery corrections in that framework Witten 2026.

These works do not yet share identical hypotheses, algebras, dressings, or error metrics. The adversarial result is therefore a live incompatibility to resolve, not a consensus verdict for or against all finite-$N$ holographic QEC.

Regime, evidence ceiling, and handoff

Track $G_N/L^{d-1}\sim N^{-2}$, $S_{\mathrm{BH}}\sim N^2$, code dimension $e^{\alpha S}$, $E_*$, the gap to a competing QES, $\alpha'/L^2$, KK and loop corrections, and errors nonperturbative in $1/N$. “Exact to all perturbative orders” permits $e^{-cN^2}$ corrections; whether those are negligible depends on the task and code size.

The evidence ceiling is an approximate channel theorem or model calculation with a declared norm. No disputed finite-$N$ shared logical algebra is reported as settled. Continue to Non-Isometric Encoding Proposals for maps with kernels and to QEC Evidence, Current Disputes, and Status for the dated comparison.

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

• Cao, C., Cheng, G., Karthikeyan, K., Li, C., and Preskill, J. (2026), “State-Dependent Geometries from Magic-Enriched Quantum Codes,” preprint, revised June 2026. arXiv:2603.13475.
• Hayden, P., and Penington, G. (2019), “Learning the Alpha-Bits of Black Holes,” Journal of High Energy Physics 2019(12), 007. DOI; arXiv:1807.06041.
• Kretschmann, D., Schlingemann, D., and Werner, R. F. (2008), “The Information-Disturbance Tradeoff and the Continuity of Stinespring’s Representation,” IEEE Transactions on Information Theory 54, 1708–1717. DOI; arXiv:quant-ph/0605009.
• Terashima, S. (2026), “Entanglement Wedge Reconstruction without Holographic Quantum Error Correction,” preprint. arXiv:2607.08684.
• Witten, E. (2026), “A Note on Corrections to Entanglement Wedge Reconstruction,” preprint, revised June 2026. arXiv:2606.18639.