A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes
Moustafa A. Radwan
In the model of approximate holographic codes introduced by Cao, Cheng, Karthikeyan, Li, and Preskill and analyzed perturbatively by Witten, the error in entanglement wedge reconstruction is smaller than the gravitational backreaction by a factor 1/d_2^2 at second order in the skewing parameter, where log d_2 plays the role of the area term. It was recently observed numerically that this hierarchy fails at fourth order for non-degenerate bulk spectra, through a coefficient growing as d_2^4. In the present work a closed form for this coefficient is proved: writing p = (p_1, ..., p_d1) for the Schmidt spectrum of the bulk state, the leading fourth-order term of the mean reconstruction error is epsilon^4 sigma_W^4 alpha(p) d_2^4, where alpha(p) = (1/2) d_1^2 (sum_i 1/p_i - d_1^2) coincides with (1/2) d_1^4 chi^2(u || p), the Neyman chi-squared divergence of the bulk spectrum from the uniform distribution u = (1/d_1, ..., 1/d_1). The mechanism is identified exactly: the ensemble mean of the second-order state perturbation is a decoherence drift of magnitude proportional to d_2^2 directed toward maximal mixing, and alpha is the Kubo-Mori quadratic form of this drift. The spectral-degeneracy protection of the hierarchy observed numerically is thereby explained as the equality case of Jensen's inequality. The formula, conjectured from five calibration spectra, was subsequently confirmed by a zero-parameter blind prediction at a previously unmeasured configuration (d_1 = 4, a new spectrum), which it matched to 0.08 per cent (-0.13 sigma). This is a corrected version of the preprint DOI 10.13140/RG.2.2.14852.69768. Three presentational defects are corrected and no result, proof, or measured value changes. First, the figure committed in advance for the blind test was 601,378, whereas the closed form proved in the paper gives 15552 alpha = 601,344 exactly; the committed figure corresponds to a rounded alpha = 38.6689 and exceeds the exact one by 34, a relative excess of 5.7e-5. Both figures are now stated with the deviation of the measurement from each, and the test is passed against either. Second, the fourth calibration spectrum was printed as (0.453, 0.333, 0.213), which does not sum to one; it is now given as the exact spectrum (34, 25, 16)/75 that was simulated, for which the stated alpha = 4.0202 is correct. Third, the blind test is described as zero-parameter rather than registered: the prediction uses no fitted quantity of any kind, but it was not deposited with a third party, so no external timestamp is claimed. The corrections are listed in the "Note on this version" on the first page. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21223051.
•Oct 24, 2026•0 citations