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.
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.
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.
The monogamy of mutual information (MMI) is a quantum entropy inequality that enforces the non-positivity of tripartite information. We investigate the failure of MMI in graph states as a forbidden-subgraph phenomenon, conjecturing that every MMI-violating graph state is local-Clifford equivalent to one whose graph contains a four-star subgraph. We construct a family of star-like graphs whose states fail a specific class of MMI instances, and extend this analysis to general star topologies. Deriving adjacency matrix constraints that fix the MMI evaluation for these instances and interpreting them physically, we prove the forbidden-subgraph conjecture for this family of graphs. Finally, through an exhaustive search over graph representatives for all 8-qubit stabilizer entropy vectors, we establish that MMI failure is not reducible to the cases within our scope.
Unrefereed mathematical candidate: a faithful-Pauli converse to the even-hole, claw-free global spectrum theorem; an exact central-sector criterion; explicit all-coupling periodic Fendley blocks at N=8,9; weak prescribed-mode heredity; and a repaired three-mode parity-completeness argument. Written proofs, exact computer-algebra certificates for 107 listed graphs, and numerical census evidence have separate assurance boundaries. Producer replay and a separately implemented model-review checker are not external human peer review, unaffiliated reproduction, formal verification or exhaustive novelty. The remaining 1621 numerical candidates are not all-coupling certified. Original prose/data CC0-1.0; original code MIT; third-party exclusions in LICENSES.md.
We study particle-loss resistant entanglement within the framework of stabilizer and graph states. A pure state is m-resistant if it remains entangled after the loss of any m particles and becomes fully separable after the loss of any m+1 particles. The smallest previously unresolved qubit case was the existence of a five-qubit 1-resistant pure state, which is resolved here by the five-cycle graph state ∣C5⟩. A stabilizer-subgroup method is also developed for verifying m-resistance in graph states, using local stabilizers to certify full separability and exact negative partial transpose~(NPT) witnesses to certify entanglement.
Applying this to all graph states associated with non-isomorphic graphs on five, six, and seven vertices, we obtain a graph state classification up to local Clifford equivalence, which also classifies stabilizer states
up to local Clifford equivalence. Thus, the five-qubit 1-resistant stabilizer states are exactly the local Clifford class of C5. Six-qubit 2-resistant stabilizer states exist in three distinct local Clifford classes, whereas no seven-qubit stabilizer state is m-resistant for any nonzero admissible m. Finally, we prove that the cycle graph states ∣CN⟩ with N≥7 are not m-resistant for any 0≤m≤N−2.
Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.
This preprint studies a structural failure mode in variational quantum mereology using the compact generalized-bipartition construction of Kabernik, Pollack, and Singh (KPS) as a benchmark. KPS previously reported non-uniqueness of the selected low-field structures, including six distinct state-reduction maps for the N=3 example. The contribution of this work is not the discovery of that degeneracy, but an explanation of its structural origin and its physical consequences.For finite open Ising chains, within the fixed computational-basis compact family, the diagonal Ising interaction does not contribute to the inter-row part of the short-time linear-entropy selector. As a result, the selector factorizes into a positive field-dependent normalization multiplied by a graph functional that is independent of the interaction ordering. This interaction blindness produces a criterion-symmetry surplus: the selector has a larger permutation symmetry than the discrete symmetry of the physical open-chain Hamiltonian.The mechanism is derived analytically, while its realization as physical non-identification is exhaustively certified for N=3, N=4, and N=5. The complete exact-selected sets contain 6, 48, and 120 compact maps, respectively, which split into 2, 12, and 30 discrete Hamiltonian-symmetry classes. An independent reduced-state information diagnostic varies across these classes, showing that structures considered equivalent by the selector can retain different physical information.For N=5, the exact selector optimum is 9/140. The full selected set contains 120 maps and 30 discrete Hamiltonian-symmetry classes, reproduced by independent exhaustive computational implementations. The accompanying reproducibility archive includes the manuscript, Supplemental Material, numerical certificates, symmetry audits, independent verification scripts, integrity manifests, and package-level reproducibility checks.An N=6 computational certificate is also included as a boundary check. It improves the certified loss bound to LOSS >= 51250 and gives the bracket 1232336 <= Z_F <= 1288670. This does not constitute closure of the exact N=6 optimum and is not used to claim an asymptotic scaling law.