{"paper":{"title":"Combining moment matrices, symmetric extension, and Lov\\'asz theta: $\\Phi_{\\text{E8}}$ is entangled","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The 14-qubit state Φ_E8 is entangled, certified by an explicit witness from a semidefinite program combining moment matrices and symmetric extension.","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Felix Huber, Gerard Angl\\`es Munn\\'e, J\\c{e}drzej Stempin, Santiago Llorens","submitted_at":"2026-05-13T17:54:51Z","abstract_excerpt":"We solve an open problem in entanglement theory posed by Yu et al., {\\it Nature Communications 12, 1012 (2021)}. The problem is to show, via an entanglement witness, that the $14$-qubit state $\\Phi_{\\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $\\Phi_{\\text{E8}}$ is entangled. The proof has the form of a rational infeasibility certificate for a semidefinite program, yielding an explicit entanglement witness. Our approach unifies and extends several earlier methods that involve the Lov\\'asz theta number of th"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We solve an open problem in entanglement theory posed by Yu et al., Nature Communications 12, 1012 (2021). The problem is to show, via an entanglement witness, that the 14-qubit state Φ_E8 is entangled.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the chosen combination of moment matrices and symmetric extension produces a tight enough relaxation whose infeasibility certificate remains valid for the full (unrelaxed) entanglement detection problem without introducing false positives.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The 14-qubit state Φ_E8 is entangled, certified by an explicit entanglement witness obtained as a rational infeasibility certificate for a semidefinite program that combines moment matrices with symmetric extension.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The 14-qubit state Φ_E8 is entangled, certified by an explicit witness from a semidefinite program combining moment matrices and symmetric extension.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0a3ce74bacb5a08f2359e1040aba4baa2e17bdc25dbc09b353e0c6fdfaef37ff"},"source":{"id":"2605.13832","kind":"arxiv","version":1},"verdict":{"id":"e094feff-93c9-4669-8c3f-c888a725a24c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:41:12.007142Z","strongest_claim":"We solve an open problem in entanglement theory posed by Yu et al., Nature Communications 12, 1012 (2021). The problem is to show, via an entanglement witness, that the 14-qubit state Φ_E8 is entangled.","one_line_summary":"The 14-qubit state Φ_E8 is entangled, certified by an explicit entanglement witness obtained as a rational infeasibility certificate for a semidefinite program that combines moment matrices with symmetric extension.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the chosen combination of moment matrices and symmetric extension produces a tight enough relaxation whose infeasibility certificate remains valid for the full (unrelaxed) entanglement detection problem without introducing false positives.","pith_extraction_headline":"The 14-qubit state Φ_E8 is entangled, certified by an explicit witness from a semidefinite program combining moment matrices and symmetric extension."},"references":{"count":45,"sample":[{"doi":"","year":null,"title":"(E8) in the arXiv version Ref","work_id":"6efe82c1-d171-44d2-a79d-d830cf8f1629","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Symmetric extension:","work_id":"d942e76c-b13c-40d5-b9ba-30968d322142","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Moment matrix: ΦE8 ? = P i piµ⊗2 i Φ(3) E8 ? = P i piµ⊗3 i  ⟨Ea⟩⟨E† b ⟩⟨E† aEb⟩   ⪰0 ∄ ΦE8 is entangled","work_id":"ddcea804-d54e-41f2-9aaf-11b057ad3016","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Combining moment matrices, symmetric extension, and Lov\\'asz theta: $\\Phi_{\\text{E8}}$ is entangled","work_id":"e60aa65a-5acf-42c0-afb3-40cd7f3cd926","ref_index":4,"cited_arxiv_id":"2605.13832","is_internal_anchor":true},{"doi":"","year":null,"title":"permutes the tensor factors byS n, and","work_id":"37cbb0e8-668b-4169-9b4d-c0332338adf7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":45,"snapshot_sha256":"ad5ab2553e7f987bcc6b5294a1a7ecb7d526905e96bc3535e4eda94fe6633222","internal_anchors":2},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}