A four-qubit counterexample to first-excited parity in phase-shifted EPR-like Hamiltonians

Shunpei Shimizu

Published Oct 10, 2026• 0 citations

Abstract

This source-only preprint gives a four-qubit counterexample to the arbitrary-phase extension of the first-excited parity conjecture discussed in Wong, Bravyi, Gosset and Liu, Lee-Yang tensors and Hamiltonian complexity, arXiv:2602.03605v1, Section 5.6. For an unweighted four-cycle with phase pi on one edge and a common parameter 0 <= s < 1, it derives the complete 16-dimensional spectrum and an exact parity threshold. The rational value s = 2/3 is an explicit witness; a specified fixed-witness robustness interval is also proved. The original zero-phase conjecture and the general all-graph gap problem remain open. Ground-state uniqueness and the non-strict gap bound are known inputs, explicitly credited to Wei, arXiv:2609.26802v2, Corollary 5.8 and Section 5.4, and Bravyi, Gosset, Liu and Wong, arXiv:2607.14401v1, Theorem 1. The concrete phase counterexample, complete spectrum and threshold are distinguished from these known results. The primary-source comparison is dated and bounded; worldwide priority is not certified. Read the unchanged PHASE_EPR_FINAL_MANUSCRIPT_v0.1.3_R2_20261010.tex or .md first, together with the current public source/verification supplement and PUBLIC_RELEASE_NOTE. Older versions and administrative private-draft wording, if present as provenance, describe historical preparation stages and do not define this public version or its reviewed scope. Auxiliary six-qubit exploration is outside this paper's completed claim. Shunpei Shimizu initiated and directed this personal independent project, arranged verification, and coordinated the materials. No university or institutional affiliation is declared. AI systems performed research exploration, proof construction, computational checks, analytical review and manuscript preparation. The reviews are internal AI analyses; independent human verification of every proof, specialist human peer review and complete proof-assistant verification are not claimed. No PDF of editorial revision R2 has been generated or visually verified: the available compiler failed during environment setup. Editable TeX, a readable Markdown manuscript and verification materials are provided. This is a limited, unrefereed research preprint, released under Creative Commons Attribution 4.0.

DOI: 10.5281/zenodo.23222980

Related papers

An infinite family of non-star trees admitting Laplacian peak state transfer

Shunpei Shimizu

Unrefereed AI-assisted preprint, version 0.2.2, 9 October 2026. No author affiliation is given. For each m >= 1, set q = m^2 + m + 1. Attach q intermediate vertices to a root and q - 1 leaves to each intermediate vertex. This produces infinitely many pairwise nonisomorphic non-star trees of order q^2 + 1, with root Laplacian spectral support {0, m^2 + 1, (m + 1)^2 + 1}. The symbolic argument establishes Laplacian peak state transfer at time pi from the root to an intermediate vertex for odd m and to a leaf for even m, addressing Open Problem 7.1 of Coutinho, Guo and Schmeits, arXiv:2505.11986v4. The ten-vertex m = 1 member is their known Figure 10 example. Phase alignment is an application of their Lemma 5.2 and Corollary 5.3; the general peak-time classification follows their Theorem 5.4 and Corollary 5.5. The contribution presented here is the explicit infinite family and its spectral coefficients, targets and gcd calculation. For the specified pairs, all positive peak times are odd multiples of pi/g, where g = 5 when m = 2 mod 5 and otherwise g = 1. Novelty and priority are not certified. Peak transfer means attaining the mutual-projector absolute-sum bound, not unit amplitude. The probabilities tend to zero along both parity subsequences. This does not establish perfect transfer, efficient long-distance quantum communication or a classification of all trees. Exact standard-library reruns check actual full-tree Laplacians for m = 1..5, quotients for m = 1..1000, and gcd/phase regressions for m = 1..10000. Those finite checks support, not replace, the symbolic all-parameter proof and the analytic all-real-time Bezout argument. No complete Lean or other proof-assistant verification is supplied. 本研究は、清水竣平が個人の自主研究として企画・進行した、AIを活用した数学探索です。清水は探索方針の指示、検証の手配、成果の取りまとめを担当しました。証明の構成、計算による検証、原稿作成の大部分にはAIを使用しています。検証方法とその範囲は本文に記載しており、人間の専門家による査読は未実施です。 This work is an AI-assisted mathematical exploration initiated and coordinated by Shunpei Shimizu as a personal independent research project. Shimizu directed the exploration, arranged verification, and compiled the results. AI systems performed most proof construction, computational checking, and manuscript preparation. The verification methods and their scope are documented in the manuscript. The work has not received specialist human peer review. The submitter has not independently certified every mathematical step. Separate AI analyses and implementations are not human refereeing. No cited-author or institutional endorsement is implied. Read peak_trees_en_v0.2.2.tex first. The current package is Quantum_peak_trees_publication_v0.2.2_20261009.zip; QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md explains its contents. Version 0.2.2 removes author affiliation; mathematical text, equations, statements, proofs and verification programs are unchanged from version 0.2.1. The current package omits old manuscript sources and old full ZIPs. Earlier verification records retain their original dates; they are not human peer review. The built-in LaTeX compiler failed at Windows environment setup before TeX diagnostics. No PDF has been generated or visually verified; editable TeX and exact-check code are supplied. Documentation clarification, 10 October 2026. The retained README_EN.txt and README_JA.md are historical documentation; the corresponding preserved v0.1.0 copies describe the 8 October 2026 working draft. QUANTUM_TREES_README_PUBLICATION_v0.2.1_EN.md describes the superseded v0.2.1 package. Their source names, payload inventories, draft-status statements and no-Publish instructions are dated history, not the current record state or operative instructions. Independent checks in those documents mean separate AI analyses or implementations, not human refereeing. The current entrypoints are peak_trees_en_v0.2.2.tex, Quantum_peak_trees_publication_v0.2.2_20261009.zip and QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md. Refer to the current TeX for the argument where historical checker comments mention an absent review.md. This notice makes no change to the mathematical manuscript or verification programs. Publication packaging clarification, 10 October 2026. In addition to the three historical READMEs identified above, the two QUANTUM_TREES_ATTRIBUTION_AI_ROLE_ADDENDUM files and novelty_attribution_math_monitor_review_20261009.zip are dated supplementary review records. The r1 addendum and the current v0.2.2 manuscript give the operative author and AI contribution disclosure. The novelty archive also compares other manuscripts; those comparisons are context, not additional results of this Quantum record. Earlier private-save and no-publication instructions describe past workflow only. Current scientific claims are the explicitly stated tree-family construction and specified-pair peak times. No mathematical content, verification program or frozen historical attachment is changed by this clarification.

Oct 10, 20260 citations

Comment on “Entanglement transformation between two-qubit mixed states by LOCC” [Phys. Lett. A 373 (2009) 3610]

Iulia Ghiu

arXiv:1008.2941Physics Letters ASep 22, 20103 citations

A Quantified Open Region of Carrier Non-Identifiability in Fragment-Averaged Pointer Information: A Parameterized Interval Certificate for Four Qubits

Yoichi Tsujisawa

Correction in version 1.1 (2026-10-18) The Decimal/Fraction interval engine distributed with version 1.0 used context-sensitive Decimal unary negation and absolute value on interval endpoints. These operations can round inward, so the recorded P02G certificate in v1.0 should not be treated as a rigorously outward-rounded enclosure. Version 1.1 repairs those operations using exact copy_negate() and copy_abs(), hardens the second interval implementation's row-bound selection, and regenerates the interval reports, checksums, and reproducibility manifest. The certified rectangle, theorem scope, and all 34 mathematical environments in the manuscript remain unchanged. Repaired P02G and the separately implemented mpmath.iv P02J both pass the original theorem gates. The result remains conditional computer-assisted mathematical evidence, not formal machine verification or external peer review. Previous version (v1.0): https://doi.org/10.5281/zenodo.23438614 This unrefereed research preprint establishes a computer-assisted, quantitatively certified non-identifiability result for fragment-size-averaged Holevo information in an N=4 environment with two equiprobable pure pointer-conditioned branches. For every point in an explicitly specified closed rational rectangle of coordinate area 4 × 10^-14, two realizations of the complete averaged Holevo information curve exist: one with minimum pure-common-rest carrier size two and another with carrier size four. The proof combines a uniform parameterized contraction mapping, directed interval arithmetic with a second interval implementation, and an intermediate-value construction. Reproducibility materials include the certification programs, frozen numerical certificates, SHA-256 checksums, and a standalone 200-state numerical comparison against direct 16-dimensional partial traces. Both interval implementations share the underlying analytic Holevo reduction and do not constitute independent symbolic proofs. The conclusions are restricted to N=4, equiprobable pure branches, fragment-size-averaged Holevo information, and the specified carrier definition. They do not establish non-identifiability for arbitrary N, mixed states, unequal priors, or individual labeled-fragment information. This is an unrefereed preprint, not a formally verified or peer-reviewed mathematical theorem. Comprehensive novelty priority is not claimed. Licensing: The manuscript and original documentation/data are provided under CC BY 4.0 where applicable. Original Python research code is provided under the MIT License. The accompanying reproducibility archive specifies licenses on a per-file basis. AI assistance disclosure: OpenAI ChatGPT assisted with mathematical analysis, Python implementation, numerical and interval verification, manuscript preparation, and quality assurance. Claude contributed critical assessment and adversarial review assistance. Neither AI system is a coauthor or an external scientific referee. The human author, Yoichi Tsujisawa, assumes responsibility for the research and its publication.

Oct 17, 20261 citation

Comment on “Multipartite Entanglement in Four-qubit Graph States”

Saeed Haddadi

arXiv:1701.08758International Journal of Theoretical PhysicsJun 9, 201710 citations

Effective Hamiltonian for two interacting double-dot exchange-only qubits and their controlled-NOT operations

Elena Ferraro, Marco De Michielis, Marco Fanciulli +1 more

arXiv:1501.05833Quantum Information ProcessingOct 30, 201420 citations

A forbidden-subgraph theorem for graph-state monogamy

Ya Zhang

Public mathematical preprint by Yongxian Zhang. We prove Fuentes–Keeler–Munizzi–Pollack, arXiv:2511.19585v1, Conjecture 1: for every finite qubit graph state, positive tripartite information implies that a locally equivalent graph contains an induced four-vertex claw. The proof gives the complete claw-vertex-minor-free component classification by K1, K2, P3, P4, C5 and the six-vertex W5, and the sharp connected threshold seven. The unbounded step is a connected-graph induction; the finite step checks all 120 nonempty one-vertex extensions with independent exact checkers. The certificate package verifies all 120 extensions, 1,511 orbit transitions and 5,460 labeled four-block assignments. These finite checks supplement the written entropy and induction proof; the theorem is not claimed as a complete Lean formalization. No external human peer review or historical-priority claim is made. Research and manuscript preparation used AI assistance; no external funding. Original material is reserved unless a file states another license. DOI registration does not certify correctness, novelty, or priority.

Oct 10, 20260 citations