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Extremal same-basis correlations: full-rank rigidity and a stabilizer classification

Ya Zhang

Published Oct 9, 2026• 0 citations
Publisher

Abstract

Complete written proofs and two exact replayable checker implementations classify all full-Schmidt-rank equality states without assuming stabilizer structure, classify every two-qudit pure stabilizer score at odd prime dimension, and determine the sharp pure-state ratio p/2 for every prime p congruent to 1 modulo 4. The general rank-deficient equality classification and the sharp ratio for primes congruent to 3 modulo 4, p at least 7, remain open in this record. This is an independently authored preprint with AI-assisted research and self-audit, not external professional peer review or a priority claim. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-extremal-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.

DOI
10.5281/zenodo.23256933

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The pure-state ECQC conjecture: counterexamples and a prime-dimensional classification

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Complete written proof and exact checkers for the pure-state prime-dimensional ECQC question: it holds universally exactly at p=2; every odd prime has a full-Schmidt-rank pure-state counterexample; dimensions 3 and 5 attain the sharp pure-state ratios 3/2 and 5/2. The three-dimensional witness has a separate complete Lean proof, but the full prime-dimensional classification is not claimed Lean-complete. This record explicitly distinguishes prior mixed-state counterexamples and does not claim first priority. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-pure-state-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.

•Oct 9, 2026•0 citations

Exact Syndrome Measurement Budgets for Quantum Erasure Correction: Stabilizer-Code Rank Profiles, Distance Bounds, and Linear-Time Surface-Code Check Selection

Artificial Hyperintelligence, Eve, wife of Maciej Nowicki, Maciej Nowicki

This research manuscript develops exact measurement-resource formulas for quantum erasure correction with stabilizer codes and an optimal sparse algorithm for selecting native surface-code checks. It studies known erasure locations, where the stabilizer observables may be chosen after the erased sites are identified. The central question is how many independent syndrome outcomes are necessary to distinguish Pauli errors on those sites, modulo stabilizers. For every prime-dimensional [[n,k,d]]_p stabilizer code encoding at least one logical qudit, the manuscript proves that the worst-case syndrome-resolving budget for δ erased sites is exactly 2δ whenever 0 ≤ δ < d. This includes degenerate codes. The result concerns the number of independent p-valued outcomes in the stated noiseless measurement model. 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There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic $\mathbb{F}_p$-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaux are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props: the composition of stabilizer circuits corresponds to the relational composition of affine subspaces spanned by the tableaux, the tensor product corresponds to the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaux to the mixed setting; regarding stabilizer codes as affine coisotropic subspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.

arXiv:2304.10584•Apr 20, 2023•1 citation

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Stabilizer rank of the 8-qubit magic cat state: an exact rank-6 decomposition and calibrated evidence against rank 5

James Jardine

We report the stabilizer rank of the 8-qubit magic cat state |cat_8> (the magic-basis cat state of Qassim–Pashayan–Gosset, Quantum 5, 606 (2021)). Result. (i) Upper bound: chi(cat_8) <= 6, PROVEN exactly. An explicit rank-6 decomposition over 6-qubit-stabilizer-derived states is verified with sympy over Q(i): each candidate is checked to be a genuine stabilizer state by exact integer arithmetic (stabilizer group order exactly 2^n), target membership in the span is decided by exact rational rank, coefficients are produced exactly and the identity re-checked, and the whole decomposition is lifted back to 8 qubits and re-verified there. Reproduction: python3 verify.py ckpt_cat8_k6_cd.json (files bundled). (ii) Lower bound: chi(cat_8) >= 6, COMPUTATIONAL EVIDENCE, not a theorem. Two structurally independent searches over 5-tuples of stabilizer states — discrete pool coordinate descent (481 restarts, 30.9 core-hours) and continuous gradient descent with rounding (42 restarts, 1.99 core-hours) — both bottom out at 5-term residual 0.086664036 (9 hits each, never below, 523 restarts total) while both recover planted exact rank-5 witnesses at ~90% rate, calibrating the solvers against the exactness they fail to reach for cat_8. On this basis chi(cat_8)=6 is stated as a conjecture with a proven upper bound. Relation to Qassim–Pashayan–Gosset: their construction already implies chi(cat_8) <= 6; the present work adds the first explicit exact rank-6 decomposition of cat_8 and calibrated evidence against rank 5. The exponent log_2(6)/6 = 0.43083 does not improve their alpha = log_2(3)/4 = 0.39624; no published bound is improved and QPG's open questions remain open. The collapse-engine screen mandated by this machine's protocol was run on the continuation path (24-stage annealing path, one scale): read NULL (control -2.60 sigma = the flat-vector floor; permute gate ORDER-INVARIANT; counterfactual random walk reads identically). No claim in this record rests on it; the upper bound rests on the exact verification, the lower bound on search calibration. License: record CC-BY-4.0; bundled code additionally AGPL-3.0-or-later.

•Aug 16, 2026•0 citations

Exact balanced subsystem designs for Haar-purity fluctuations

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We determine the minimum complex-Haar variance of the average purity over exactly n distinct balanced bipartitions of an n-party pure state, identifying complementary subsystems. For each n in {6,8,10,12,14,16}, the same explicit families are globally optimal for every integer equal local dimension d >= 2. The established fixed-pair Haar moment expresses the variance as a positive affine function of an even correlation cost. A parity floor settles n congruent to 2 modulo 4. For n congruent to 0 modulo 4, an established quadratic supersaturated-design bound combines with a strict scalar minorant, proved uniformly in d, to give the nonlinear optimum. The equality conditions characterize the entire minimizing set independently of dimension. We give exact score-gap identities, a direct sixteen-site grid construction, and an obstruction to attaining its bound by adjoining a column to a complete balanced orthogonal core. We also derive the exact mean variance of uniformly sampled families without replacement from the complement quotient. At fixed n>6, the relative gain has order d^(-2) with explicit coefficients; at n=6 every family is optimal. Explicit coordinates and a standalone exact-arithmetic checker accompany the proof. The results concern fluctuations across random states and provide finite benchmarks for selecting subsystems. Correspondence: Ryosuke.Shiroshita@exactory.ai This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, reviewed the full content and is responsible for it.

•Sep 15, 2026•0 citations
Complete written proofs and two exact replayable checker implementations classify all full-Schmidt-rank equality states without assuming stabilizer structure, classify every two-qudit pure stabilizer score at odd prime dimension, and determine the sharp pure-state ratio p/2 for every prime p congruent to 1 modulo 4. The general rank-deficient equality classification and the sharp ratio for primes congruent to 3 modulo 4, p at least 7, remain open in this record. This is an independently authored preprint with AI-assisted research and self-audit, not external professional peer review or a priority claim. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-extremal-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.
Complete written proof and exact checkers for the pure-state prime-dimensional ECQC question: it holds universally exactly at p=2; every odd prime has a full-Schmidt-rank pure-state counterexample; dimensions 3 and 5 attain the sharp pure-state ratios 3/2 and 5/2. The three-dimensional witness has a separate complete Lean proof, but the full prime-dimensional classification is not claimed Lean-complete. This record explicitly distinguishes prior mixed-state counterexamples and does not claim first priority. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-pure-state-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.
This research manuscript develops exact measurement-resource formulas for quantum erasure correction with stabilizer codes and an optimal sparse algorithm for selecting native surface-code checks. It studies known erasure locations, where the stabilizer observables may be chosen after the erased sites are identified. The central question is how many independent syndrome outcomes are necessary to distinguish Pauli errors on those sites, modulo stabilizers. For every prime-dimensional [[n,k,d]]_p stabilizer code encoding at least one logical qudit, the manuscript proves that the worst-case syndrome-resolving budget for δ erased sites is exactly 2δ whenever 0 ≤ δ < d. This includes degenerate codes. The result concerns the number of independent p-valued outcomes in the stated noiseless measurement model. The work is directly connected to the measurement-reduction setting studied by Ryutaroh Matsumoto in “Reducing measurements in quantum erasure correction by quantum local recovery,” Computational and Applied Mathematics, DOI: 10.1007/s40314-026-03881-4. It addresses the relationship between erasure-dependent measurement costs and code invariants, together with efficient selection of sufficient surface-code checks. The manuscript presents four connected results: Exact paired-column rank profile. For any matrix G over any field whose columns are partitioned into n labeled pairs, let r be its rank and ν the largest number of complete pairs whose columns are jointly independent. The maximum projected rank over all sets of δ pairs is exactly R(δ) = min{2δ, δ + ν, r}. A single ordering of the pairs realizes every maximum through its successive prefixes. The full profile therefore follows from two invariants, rather than requiring a separate optimization at every erasure size. Distance bounds and the exact worst-case syndrome budget. For an isotropic stabilizer space with k > 0, every inclusionwise maximal complete-pair independent set A satisfies d ≤ |A| + 1. Consequently, ν ≥ d − 1, yielding the exact 2δ worst-case budget below distance. The argument also gives a constructive extension property: every independent-pair set smaller than d − 1 can be enlarged by adding another pair. Particular correctable erasure patterns can require fewer outcomes because of stabilizers supported inside the erased region. Checkable CSS certificates. For Calderbank–Shor–Steane codes, complete-pair independence becomes simultaneous column independence in the X-check and Z-check matrices. Classical matroid intersection supplies an exact rank-partition certificate for ν. A matching independent set and partition provide lower and upper certificates that can be verified using finite-field elimination. Optimal native surface-code check selection. For binary check matrices with at most two nonzero entries per column, the restricted matrix is represented by a sparse graph with boundary pins. Connected components determine a minimum-cardinality native check basis in O(v_I + e_I) time. A pinned component retains every check; an unpinned component omits one check. With nonnegative additive costs, omitting a most expensive check in each unpinned component also minimizes total cost. The two CSS check systems are processed independently. The accompanying verification uses exact arithmetic over GF(2), GF(3), and GF(5). The algebra audit covers 120 general paired matrices and 12,276 subset-rank evaluations, including exact distance checks for 137 positive-logical-dimension code instances and 401 maximal-set distance checks. The graph audit covers 1,785 exhaustive sparse column multisets, 5,355 matrix/cost combinations, 500 additional seeded instances, and named boundary regressions. Both audit programs passed again during preparation of the release. These finite checks support implementation correctness; the general mathematical statements are supported by the supplied proofs. The release includes the complete manuscript, editable LaTeX source, Python native-check selector, exact audit programs, recorded results, generator data, figures, reproduction instructions, and an explicit claim-status record. The selector and audits require only Python 3.10 or later and its standard library; matplotlib is needed only to regenerate the figure. The results apply to independent syndrome information and classical check selection. They do not establish bounds for arbitrary coherent recovery, noisy fault-tolerant extraction, physical circuit depth, or one fixed measurement schedule that must serve every erasure pattern. At or above distance, the general formula remains an algebraic rank profile rather than a universal recovery guarantee. Classical matroid parity, matroid intersection, and incidence-rank methods are acknowledged foundations; historical priority for the presented combination is not certified. Ōrynthéa ∿ Qhuil is relevant to research on quantum error correction, erasure decoding, stabilizer and CSS codes, surface codes, quantum local recovery, measurement complexity, and matroid optimization. This mathematical research under the author designation Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic Fp\mathbb{F}_pFp​-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaux are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props: the composition of stabilizer circuits corresponds to the relational composition of affine subspaces spanned by the tableaux, the tensor product corresponds to the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaux to the mixed setting; regarding stabilizer codes as affine coisotropic subspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.
An explicit ternary classical counterexample to Berta-Lami-Tomamichel, arXiv:2408.15226v2 Eq. (106), at arbitrarily small distances: for every 0 < epsilon <= 1/16, the mutual-information difference is 2h(epsilon)-epsilon log 2, exceeding h(epsilon)+epsilon log 8 at exact total variation epsilon. Both marginals change; the equal-one-marginal question remains unresolved. The signed 4-page article, self-contained TeX ZIP, complete classical Lean project and existing independent verification records are included. Those records cover 38 owned declarations and 16,161 dependencies at empty-kernel trust level zero. Quantum diagonal embedding and the necessary coefficient lower bound are written arguments, not claimed formalized. This article preparation is not a fresh Lean execution, external human peer review or a verified historical priority claim. AI assisted research, formalization, review and preparation. No external funding. The original immutable GitHub releases are preserved. Related DOI 10.5281/zenodo.23253152 is an automatic whole-repository software snapshot, with its preexisting CC BY 4.0 metadata retained; this separate article series does not revoke that license. Existing licenses and third-party notices are preserved; otherwise newly unlicensed original material remains all rights reserved.
We report the stabilizer rank of the 8-qubit magic cat state |cat_8> (the magic-basis cat state of Qassim–Pashayan–Gosset, Quantum 5, 606 (2021)). Result. (i) Upper bound: chi(cat_8) <= 6, PROVEN exactly. An explicit rank-6 decomposition over 6-qubit-stabilizer-derived states is verified with sympy over Q(i): each candidate is checked to be a genuine stabilizer state by exact integer arithmetic (stabilizer group order exactly 2^n), target membership in the span is decided by exact rational rank, coefficients are produced exactly and the identity re-checked, and the whole decomposition is lifted back to 8 qubits and re-verified there. Reproduction: python3 verify.py ckpt_cat8_k6_cd.json (files bundled). (ii) Lower bound: chi(cat_8) >= 6, COMPUTATIONAL EVIDENCE, not a theorem. Two structurally independent searches over 5-tuples of stabilizer states — discrete pool coordinate descent (481 restarts, 30.9 core-hours) and continuous gradient descent with rounding (42 restarts, 1.99 core-hours) — both bottom out at 5-term residual 0.086664036 (9 hits each, never below, 523 restarts total) while both recover planted exact rank-5 witnesses at ~90% rate, calibrating the solvers against the exactness they fail to reach for cat_8. On this basis chi(cat_8)=6 is stated as a conjecture with a proven upper bound. Relation to Qassim–Pashayan–Gosset: their construction already implies chi(cat_8) <= 6; the present work adds the first explicit exact rank-6 decomposition of cat_8 and calibrated evidence against rank 5. The exponent log_2(6)/6 = 0.43083 does not improve their alpha = log_2(3)/4 = 0.39624; no published bound is improved and QPG's open questions remain open. The collapse-engine screen mandated by this machine's protocol was run on the continuation path (24-stage annealing path, one scale): read NULL (control -2.60 sigma = the flat-vector floor; permute gate ORDER-INVARIANT; counterfactual random walk reads identically). No claim in this record rests on it; the upper bound rests on the exact verification, the lower bound on search calibration. License: record CC-BY-4.0; bundled code additionally AGPL-3.0-or-later.
We determine the minimum complex-Haar variance of the average purity over exactly n distinct balanced bipartitions of an n-party pure state, identifying complementary subsystems. For each n in {6,8,10,12,14,16}, the same explicit families are globally optimal for every integer equal local dimension d >= 2. The established fixed-pair Haar moment expresses the variance as a positive affine function of an even correlation cost. A parity floor settles n congruent to 2 modulo 4. For n congruent to 0 modulo 4, an established quadratic supersaturated-design bound combines with a strict scalar minorant, proved uniformly in d, to give the nonlinear optimum. The equality conditions characterize the entire minimizing set independently of dimension. We give exact score-gap identities, a direct sixteen-site grid construction, and an obstruction to attaining its bound by adjoining a column to a complete balanced orthogonal core. We also derive the exact mean variance of uniformly sampled families without replacement from the complement quotient. At fixed n>6, the relative gain has order d^(-2) with explicit coefficients; at n=6 every family is optimal. Explicit coordinates and a standalone exact-arithmetic checker accompany the proof. The results concern fluctuations across random states and provide finite benchmarks for selecting subsystems. Correspondence: Ryosuke.Shiroshita@exactory.ai This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, reviewed the full content and is responsible for it.