{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation", "title": "Supporting affine functionals for Entanglement of Formation", "canonical_statement": "The paper disproves the assumption that finite-dimensionality and the convex-roof structure of Entanglement of Formation guarantee a global supporting affine functional at every bipartite state. It gives an explicit degenerate two-qubit state $\\rho$ for which no Hermitian $\\Lambda_\\rho$ satisfies both $E_F(\\rho)=\\mathrm{Tr}\\Lambda_\\rho\\rho$ and $E_F(\\sigma)\\geq\\mathrm{Tr}\\Lambda_\\rho\\sigma$ for every state $\\sigma$. The construction uses the equivalence between existence of such a functional and Lipschitz lower semicontinuity of $E_F$, together with Wootters’ formula.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:event", "problem_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation", "type": "disproved", "status": "resolved", "occurred_at": "2026-08-27T00:00:00.000Z", "title": "arXiv", "summary": "For two qubits, the paper considers\n\n$\\rho=\\frac12|\\Phi^+\\rangle\\langle\\Phi^+|+\\frac12|01\\rangle\\langle01|$,\n\nwhere $|\\Phi^+\\rangle=(|00\\rangle+|11\\rangle)/\\sqrt2$. This rank-2 state has no global supporting affine functional for Entanglement of Formation.\n\nSetting $\\rho_t=(1-t)\\rho+t|10\\rangle\\langle10|$, Wootters’ formula gives\n\n$C(\\rho_t)=\\frac12-\\sqrt{2t}+O(t)$.\n\nConsequently,\n\n$\\lim_{t\\to0^+}[E_F(\\rho)-E_F(\\rho_t)]/t=+\\infty$,\n\nso $E_F$ is not Lipschitz lower semicontinuous at $\\rho$. By the paper’s criterion, no global supporting affine functional exists there. Thus the claimed universal existence fails even for two qubits, although it remains true for nondegenerate finite-dimensional states.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:attempt" ], "method_family_ids": [ "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:method" ], "source_assertion_ids": [ "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:assertion" ] } ], "attempts": [ { "id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:attempt", "problem_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation", "model": "Claude Fable 5", "provider": "Anthropic", "attempted_at": "2026-08-27T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "From the abstract: \"We use Wootters' formula and the help of Claude Fable 5 to find a state $\\rho$ of the system $AB$ for which the latter property does not hold.\" The authors had already reduced the existence of a global supporting affine functional to Lipschitz lower semicontinuity, and knew in principle that Wootters' two-qubit formula could yield a counterexample; what they did not have was a practical way to construct one. They report the model found such a state very quickly, and it is the state of Proposition 3.\n\nAI-co-developed rather than AI-discovered, and the distinction is the tier's definition rather than a judgement call: this is a subproblem the authors formulated, inside a proof they set up, which the model solved. The surrounding theory - the equivalence criterion, the existence conditions, the Lipschitz bounds in finite and infinite dimensions - is the authors'.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.27363", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Supporting affine functionals for Entanglement of Formation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:verification", "problem_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation", "solution_event_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Site-confirmed: the counterexample was re-derived here on 28 August 2026, in exact arithmetic, from the entry's statement rather than the paper's method. Wootters' concurrence was implemented from scratch and evaluated symbolically on $\\rho=\\frac12|\\Phi^+\\rangle\\langle\\Phi^+|+\\frac12|01\\rangle\\langle01|$ and on $\\rho_t$.\n\nResults. $C(\\rho)=1/2$ exactly. $C(\\rho_t)$ has exact closed form at each rational $t$ - at $t=10^{-6}$ it is $999999/2000000-3\\sqrt{222222}/10^6$, which differs from $\\frac12-\\sqrt{2t}$ by $-4.99\\times10^{-7}$, the $O(t)$ term with coefficient about $-\\frac12$, confirming the paper's expansion. The difference quotient $[E_F(\\rho)-E_F(\\rho_t)]/t$ evaluates to $14.81$, $154.7$, $1550.9$, $15512.7$ and $155131.3$ at $t=10^{-2},10^{-4},10^{-6},10^{-8},10^{-10}$: a factor of ten per two decades, so it grows like $t^{-1/2}$ and diverges. $E_F$ is therefore not Lipschitz lower semicontinuous at $\\rho$, which by the paper's own criterion is exactly the failure claimed.\n\nWhat this does not establish. The equivalence between a global supporting affine functional and Lipschitz lower semicontinuity is the paper's, and was not checked here; nor were the further existence conditions or the infinite-dimensional bounds. The paper is an unrefereed preprint (v1, 27 August 2026, quant-ph) with no independent review.", "sources": [ { "label": "VibeMathed: Supporting affine functionals for Entanglement of Formation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2608.27363", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.27363", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Supporting affine functionals for Entanglement of Formation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation:signal:watch_only", "problem_id": "vibemathed:supporting-affine-functionals-for-entanglement-of-formation", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }