The disturbance of encoded logical information by weak coherent noise is studied at the state level, before any syndrome measurement, decoding, or recovery, for noise drawn from a centered Gaussian ensemble of Hamiltonian perturbations. Two deterministic identities are established. First, the second-order growth of the relative entropy of the reduced logical state is a Kubo-Mori-weighted sum of Knill-Laflamme violation amplitudes of the ensemble modes; it therefore vanishes identically, for every realization strength, whenever all modes have weight below the code distance. Second, the ensemble mean of the second-order logical deformation is an induced dephasing drift with the closed form -sum_mu N_ac(Pbar_mu) <Pbar_mu>_0 sigma_mu, where N_ac(Pbar_mu) counts the modes anticommuting with the bare logical Pauli Pbar_mu: distance thus removes the second-order mechanism while the drift survives as the leading corruption, entering the relative entropy at fourth order with a closed-form coefficient. For detection codes the first identity yields a deterministic, pair-resolved danger map for crosstalk. The theory was subjected to a zero-parameter blind test on the [[4,2,2]] code under XX crosstalk: the predicted six-pair map, total corruption rate, and drift matrix were committed in advance and confirmed within statistical errors, while the two nominally-zero pairs showed a small excess whose epsilon^2 scaling identifies it as the predicted quartic drift term rather than a second-order violation; its magnitude also carries a connected contribution, not obtained in closed form here. All quantities are computable by finite mode sums from the noise covariance, without sampling. This paper is posted for the first time in corrected form; it was not published previously. The corrections applied before posting concern the description of the two nominally-zero pairs of the blind test, whose measured quartic excess exceeds the computed disconnected coefficient by a factor of about three, the balance being a connected remainder that is characterized numerically here and obtained in closed form in the companion paper; and the replacement of "registered" by "zero-parameter", since the predictions use no fitted quantity but were not deposited with a third party, so no external timestamp is claimed. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21246031.
The disturbance of encoded logical information by weak coherent noise is studied at the state level, before any syndrome measurement, decoding, or recovery, for noise drawn from a centered Gaussian ensemble of Hamiltonian perturbations. Two deterministic identities are established. First, the second-order growth of the relative entropy of the reduced logical state is a Kubo-Mori-weighted sum of Knill-Laflamme violation amplitudes of the ensemble modes; it therefore vanishes identically, for every realization strength, whenever all modes have weight below the code distance. Second, the ensemble mean of the second-order logical deformation is an induced dephasing drift with the closed form -sum_mu N_ac(Pbar_mu) <Pbar_mu>_0 sigma_mu, where N_ac(Pbar_mu) counts the modes anticommuting with the bare logical Pauli Pbar_mu: distance thus removes the second-order mechanism while the drift survives as the leading corruption, entering the relative entropy at fourth order with a closed-form coefficient. For detection codes the first identity yields a deterministic, pair-resolved danger map for crosstalk. The theory was subjected to a zero-parameter blind test on the [[4,2,2]] code under XX crosstalk: the predicted six-pair map, total corruption rate, and drift matrix were committed in advance and confirmed within statistical errors, while the two nominally-zero pairs showed a small excess whose epsilon^2 scaling identifies it as the predicted quartic drift term rather than a second-order violation; its magnitude also carries a connected contribution, not obtained in closed form here. All quantities are computable by finite mode sums from the noise covariance, without sampling. This paper is posted for the first time in corrected form; it was not published previously. The corrections applied before posting concern the description of the two nominally-zero pairs of the blind test, whose measured quartic excess exceeds the computed disconnected coefficient by a factor of about three, the balance being a connected remainder that is characterized numerically here and obtained in closed form in the companion paper; and the replacement of "registered" by "zero-parameter", since the predictions use no fitted quantity but were not deposited with a third party, so no external timestamp is claimed. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21246031.
For a pure encoded state perturbed as exp(i epsilon W) with W drawn from an arbitrary centered Gaussian ensemble of Hermitian operators, the complete fourth-order coefficient of the mean relative-entropy disturbance of any reduced state is derived in closed form. The coefficient decomposes into five channels - the disconnected drift term and four connected channels - each an explicit finite sum over pairs of Karhunen-Loeve modes of the noise covariance, contracted with exact confluent divided-difference kernels of the logarithm. This closes, for arbitrary structured ensembles, the remainder that previous work in this programme controlled only for unitarily invariant ensembles: for the invariant case the mode-completeness identity reproduces Wick's theorem and the pair sums collapse to the known closed forms, while for structured ensembles all five channels are computable deterministically with no sampling, and the dominance of the drift term is governed by a computable coherence ratio rather than a universal law. Two consequences are developed. For stabilizer codes under Knill-Laflamme-protected Pauli noise the three commutator channels vanish identically and the total quartic coefficient reduces to a pure counting expression, verified rational-exactly at three configurations spanning the five-qubit, Steane, and [[4,2,2]] codes. Every closed form is certified against direct simulation, including a genuinely retrodictive test: two quartic coefficients measured and archived in an earlier study of this programme, before the present theory existed, are reproduced to 0.0 and +0.4 standard errors, with the previously unexplained connected excess of one of them given exactly by a new channel formula. This paper is posted here for the first time. It is the fifth of a series: the reconstruction-error hierarchy (DOI 10.13140/RG.2.2.19676.14723), the chi-squared divergence law (Zenodo DOI 10.5281/zenodo.23621356), the induced Lindblad drift (Zenodo DOI 10.5281/zenodo.23621898), and the Knill-Laflamme danger maps (Zenodo DOI 10.5281/zenodo.23622023). Its references point to the corrected versions of those papers. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21246031.
For a stabilizer code carrying one tracked logical qubit, perturbed by a centered Gaussian ensemble of Pauli Hamiltonian modes, the fourth-order coefficient of the mean relative-entropy disturbance of the reduced logical state was conjectured by the present author to be the pure counting expression (1/2) N_ac (N_ac + 2) <Zbar>_0^2 sum_u 1/p_u, with N_ac the number of modes anticommuting with the bare logical Zbar; it was certified rational-exactly at three configurations and reported as an exactly verified conjecture. This note proves it. Both second-order kernels are evaluated in closed form by Pauli algebra alone. The diagonal kernel is K_aa = (1 - eps_a) <Zbar>_0 sigma_z, with eps_a the commutation sign of the mode with Zbar, which yields the induced drift, its quartic coefficient, and the diagonal part of the connected channel unconditionally - using neither the Knill-Laflamme conditions nor the code distance. The off-diagonal kernel carries a sign factor f_m = 1 + eps_am eps_bm - eps_bm eta_ab - eps_am multiplying an expectation that vanishes unless the product of the two modes lies in the normalizer of the stabilizer group. This gives a necessary and sufficient vanishing criterion and, from it, the explicit hypothesis the counting law needs: 2w < d_N(S), where w is the mode weight and d_N(S) the weight of the lightest non-identity normalizer element. The hypothesis covers the two distance-three certifications, explains why the law fails on the distance-two [[4,2,2]] code, and is sufficient but not necessary - the degenerate Shor code violates it yet satisfies the law, because the sign factor annihilates the offending pairs. Every kernel identity is certified mode by mode and pair by pair: 732 modes and 1,602 pairs across four codes, with no exception. This note is posted here for the first time. It proves a conjecture stated in A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208) and resting on the stabilizer-code framework of Knill-Laflamme Violations, Induced Logical Dephasing, and Deterministic Danger Maps for Coherent Noise on Stabilizer Codes (Zenodo DOI 10.5281/zenodo.23622023). The scripts reproducing every number in the note are deposited in this record as counting_law_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only. Version 2 adds counting_law_scripts.zip, which an upload error left out of version 1. The note itself is unchanged: the PDF is byte-for-byte the file published in version 1.
Coherent Hamiltonian drifts and systematic errors can pass through quantum error correction differently than stochastic Pauli noise: they generate phase components of the logical channel that are lost under full Pauli twirling. We consider the Lindbladian dynamics of n physical qubits propagated through an effective logical channel of a fixed encoding, recovery, and decoding scheme. The work is methodological in character: standard ideas of observability and minimal realization from linear systems theory are applied to the channel \[
ω
\mapsto
Φ_t(ω)
=
\cD e^{t\cL}\cE(ω), \] where \(\cE\) encodes the logical state and \(\cD\) is the chosen recovery map with final decoding. The minimal code-visible space \(\cK_{\rm code}\) is defined as the Krylov closure of the decodable logical observables with respect to \(\cL^\dagger\); its observable factorization yields a minimal realization of the logical channel. In this form, the Knill--Laflamme conditions serve as a short-time consistency check, Pauli noise on stabilizer codes reduces to a chain on syndrome-logical classes, and the choice of recovery map acts as a spectral filter on the visible modes. For quantum memory, the same language yields an estimate of the storage time in terms of the dominant visible eigenvalues of the logical channel of the correction cycle. Particular attention is paid to the spectral signatures of coherent noise and to how these signatures depend on the chosen decoder.
A pure encoded state of an approximate quantum code is perturbed by exp(i eps W) with W drawn from a centered Gaussian ensemble, and D(eps) denotes the relative-entropy disturbance of a reduced state. This paper derives the complete fourth-order coefficient of the mean disturbance in closed form, and identifies what that coefficient is made of. The derivation has two stages, and they assume different things. The first assumes nothing about the noise beyond that it is centered and Gaussian: the coefficient splits into five channels, one disconnected drift term and four connected ones, each a finite sum over pairs of Karhunen-Loeve modes of the covariance contracted with exact confluent divided differences of the logarithm. For any concrete noise model the whole quartic disturbance is then computable deterministically, with no sampling. The second stage adds one hypothesis, that the mode family is complete and orthonormal, and evaluates every channel. The device throughout is to substitute the integral representation of the divided differences before performing the index sums, which turns each Kronecker delta into a coincidence of resolvent labels, and then to integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] of resolvent traces. No coefficient is fitted and no index contraction survives. What survives is four scalar functions of the Schmidt spectrum: Xi, chi, S_w and T. The assembled coefficient is a function of d_1, d_2 and those four alone, and four is the right number. The five functions 1, Xi, chi, T, S_w are proved linearly independent for every d_1 >= 3, analytically: a degeneration lemma confines the kernel to two dimensions and an expansion about the degenerate spectrum closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2, where the two relations that do make the set reducible are exhibited. The four are not a list that happens to close. Each is a contraction of the Kubo-Mori metric and the Amari-Chentsov skewness tensor on two operator directions, the identity and the maximal-coherence operator of the Schmidt eigenbasis; and those directions are not chosen but read off the mode two-point function, which the Gaussian ensemble fixes identically in the spectrum and for every d_1. The two are Kubo-Mori orthogonal, so chi is the squared classical Fisher length of the spectrum and S_w - chi the squared coherent length, bounded below sharply by d_1^2 (d_1 - 1). Every channel is a rational-weighted sum of metric and skewness contractions, with d_2 the only number entering from outside the geometry, and the skewness tensor enters the whole coefficient exactly once: at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term. The paper is explicit about scope. A dedicated section separates what is proved about the ensemble alone, which is everything, from what the code supplies, which is the Schmidt spectrum and the reading of log d_2 as an area term, and from where the gap lies. On a family of Pauli strings satisfying the Knill-Laflamme conditions on a stabilizer code the first-order mode objects vanish identically, so three of the five channels are not small but exactly zero, and the entire Amari-Chentsov contribution goes with them: the mechanism is a property of the ensemble, not of approximate codes in general. The step from the coefficient to a physical reconstruction-error floor requires fixing the backreaction at a chosen fraction of its Page value, a modelling decision that this paper does not make. The limitations section states four further points, including one archived measurement at d_1 = 3, d_2 = 12 which is located at a definite configuration, quantified, and left unadjudicated with the two reasons given. Every closed form is certified. The checks run from the defining index sums, from the raw five-channel sum built from the mode objects on an explicit orthonormal basis, from the geometric identifications, and, beyond the invariant ensemble, against direct simulation on structured families and stabilizer codes. Two of the latter are retrodictions: coefficients measured and archived earlier in this programme, before the present theory existed, are reproduced to +0.4 and 0.0 standard errors, with a previously unexplained connected excess given exactly by the connected-channel formula. The appendix evaluates the eighteen connected contractions on which the connected channel rests, and the deposited code derives all eighteen of them from the Wick rule and then checks symbolically that they sum to the closed form. This paper supersedes and absorbs three earlier preprints of this programme: 10.5281/zenodo.23644208, 10.5281/zenodo.23684111 and 10.5281/zenodo.23726789, together with the contraction appendix of 10.5281/zenodo.23621356. Those records remain available and are not withdrawn; readers should cite the present paper for any of their results, which are proved here in full and without reliance on unrefereed material. Version 3 corrects the presentation of the appendix table of eighteen connected contractions. Up to version 2 the nine word pairs were listed in five grouped rows, on the ground that the relations T_3 = T_1^dagger and conjugation with u <-> v relate the members of each row; those relations preserve a pair's d_2-degree but not its value, and for three pairs the grouped entry is therefore not the pair's own. The table is now listed pair by pair, the collected sum before the kernel contraction is displayed, and the deposited script derives all eighteen entries from the Wick rule rather than taking them as given. No result of the paper changes: the two matchings of a word pair are added together, and sum_uv w_uv p_u = sum_uv w_uv p_v by the symmetry of w, so the collected sum and everything downstream of it are unaffected. Version 3 also corrects the wording of the proof of the entropy-kernel lemma. The first-order term of the expansion of Tr f(sigma) is Tr[Delta log sigma_0] + Tr Delta, and it is removed by the second, linear term of the relative entropy rather than by Tr Delta = 0 as the earlier versions said; the conclusion of the lemma is unchanged and is verified in the deposited code. A fifth script, an independent review pass over the statements the other four do not cover, is added to the archive. The verification scripts that generate every entry of the certification table, together with the derivation and symbolic verification of the appendix table, are included in the attached archive under the MIT licence.
The complete fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise decomposes into five channels. Closed forms were previously available for two of them. This note evaluates the remaining three for the unitarily invariant ensemble, so that the whole coefficient is explicit for every bulk dimension d_1, every geometric dimension d_2 and every spectrum. The delta1-delta3 channel, reported in earlier work as measured but not evaluated, is A_13 = -(5/3) D (Xi - 1), with D = (d_1 d_2)^2 the total Hilbert-space dimension and Xi the Kubo-Mori spectral functional of the second-order theory. The proof uses only the two operator identities sum_a A_a^2 = D and sum_a A_a M A_a = Tr(M), and is independent of the code. The channel is negative for every spectrum, and not merely because Xi > 1: since the logarithmic mean is dominated by the arithmetic one, Xi >= d_1^2 with equality exactly at a degenerate spectrum, so A_13 <= -(5/3) D (d_1^2 - 1). The quartic commutator channel C is obtained in closed form as a contraction of the divided-difference kernel l_3 with two copies of the mode-sum tensor N, and is proved independent of d_2. The mixed channel B splits as B_1 + B_2. The remainder B_2 is the one that resisted: it is obtained here in closed form for every d_1, every d_2 and every spectrum, B_2 = 3 Xi - chi/2 - 2 + T, with chi = sum_u 1/p_u and T = sum_uvs l_2(u,v,s) p_u, where l_2 is the second confluent divided difference of the logarithm. T is a third spectral functional, and it is shown not to be a function of the other two: the best affine fit to {Xi, chi, 1} over forty interior spectra leaves worst residuals 2.86, 8.34 and 13.80 at d_1 = 3, 4, 5, against values of abs T reaching 82, 297 and 542. T is nevertheless not an alien object. In the variables m_k(t) = sum_u (p_u + t)^(-k) and h(t) = sum_u p_u/(p_u + t), the three functionals are Xi = integral h m_1 dt, chi = integral m_2 dt and T = -integral h m_1^2 dt, so T is the next member of one family. T and chi moreover scale alike, both as p^(-1), so no counting of degrees separates them; only the mode sum does. The derivation has two halves. The double mode sum Q_uvs = sum_ab (J_a)_uv (J_b)_vs (K_ab + K_ba)_su closes on the bulk spectrum alone, by four completeness identities: two that the preceding paper already prints, and two that carry an uncontracted mode operator and are what close the terms of K carrying A_a A_b Phi. Twenty-four cases are involved, and they are tabulated in the derivation note deposited in this record. The resulting divided-difference contractions then collapse by two identities proved from the integral representation of the Kubo-Mori kernels, F_4 = -d_1^2/2 and F_3 = d_1^2/2 - Xi. Three statements that could previously only be checked numerically follow as corollaries: B_2 is independent of d_2 and of the code; at a degenerate spectrum B_2 = -(1/2)(d_1^2 - 1)(d_1^2 - 4); and at d_1 = 2 it reduces to 2 (Xi - chi), that collapse being an accident of the dimension, since T there equals -(3/2) chi - w_12. Consequently the assembled coefficient is explicit at every configuration, its d_2-independent constant collapsing to 2 Xi - chi/2 - 1 + T + C. At d_1 = 2, d_2 = 2 and p = (0.7, 0.3) it reproduces a direct sampling measurement to 0.13 standard errors, at d_1 = 3 with a degenerate spectrum to 0.45, and the formula for A_13 reproduces retrodictively, to 0.55 standard errors, a measurement published before this derivation existed. Every closed form is certified against exact mode sums on an explicit orthonormal Hermitian basis of the full space, link by link rather than only end to end, and the formula for B_2 is tested on held-out spectra it never saw. An erratum to the preceding paper is also recorded. As printed, its third-order mode object L_abc = R([A_a,[A_b,[A_c,rho]]]) is anti-Hermitian, which would force A_13 to vanish identically for every ensemble, every code and every spectrum, contradicting the nonzero delta1-delta3 word measured in earlier work. The third-order deformation carries i^3, and restoring that factor removes the contradiction. That paper's own numerics evidently carried the factor, because it reports nonzero values of A_13 on two families that are not stabilizer codes; no value reported in any of the preceding papers is affected, and what requires correction is one definition as set. Two limitations are stated. Everything here is for the unitarily invariant ensemble, and none of the operator arguments survives for a structured mode family such as a Pauli basis on a stabilizer code. And a discrepancy with a published measurement at d_1 = 3, d_2 = 12 is located at a definite configuration but not adjudicated: the coefficient gives -9492 at the degenerate endpoint there, while the estimator used in this note cannot be carried to d_2 = 12 at the coupling used throughout, the Taylor control eps times the norm of W reaching 1.44. This note is posted here for the first time. It supplies the three missing channels of the five-channel decomposition of A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), corrects one printed definition in it, and evaluates in closed form the delta1-delta3 word measured in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The degeneracy measurement it compares against is reported in Breakdown and Persistence of the Reconstruction-Error Hierarchy in Approximate Holographic Codes (DOI 10.13140/RG.2.2.19676.14723). The derivation note carrying the twenty-four case tables is deposited in this record as B2_derivation_note_2026-10-27.pdf. The scripts reproducing every number in the note are deposited as closed_forms_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only.
In approximate holographic codes obtained by skewing an exact code with a Gaussian random perturbation, the accuracy of entanglement wedge reconstruction has been shown to depend strongly on the structure of the perturbation ensemble: a 1/d_2^2 error-to-backreaction hierarchy for the unitarily invariant ensemble at second order, a chi-squared-divergence floor at fourth order for generic bulk spectra, an unsuppressed second-order floor for local cross-cut perturbations, and exact protection for perturbations confined to the geometric sector. This paper unifies these behaviours. For an arbitrary centered Gaussian Hermitian ensemble, it is proved that the ensemble mean of the second-order state deformation equals the action of a Lindblad dissipator whose jump operators are the Karhunen-Loeve modes of the covariance, reduced to the code sector; and that the second-order mean relative entropies are explicit Kubo-Mori quadratic functionals of the per-mode commutator amplitudes. The reconstruction-error floor at fixed backreaction then decomposes into two mechanisms, a second-order term given by a ratio of these functionals and a quartic term given by the Kubo-Mori norm of the induced drift, from which the previously observed laws follow as corollaries, including a closed-form derivation of the coupling-fraction law that had been measured but not derived, and a new exact-protection class: any perturbation diagonal in the Schmidt basis of the bulk state leaves the reduced code state invariant to all orders. The theory is deterministic in the ensemble covariance and was subjected to a second-generation blind test: three zero-parameter predictions (a hierarchy ratio, a drift matrix, and a quartic drift coefficient) were fixed in advance for a chiral cross-cut family never previously simulated, and all three were confirmed within statistical errors. This is a corrected version of the preprint DOI 10.13140/RG.2.2.35666.93125. The word "registered" is replaced by "zero-parameter" throughout, and a footnote records the provenance of the committed predictions: they are computed from the ensemble covariance with no fitted quantity and were fixed before the corresponding measurements were run, but were not deposited with a third party, so no external timestamp is claimed. No result, proof, prediction, or measured value changes. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21230383.