Supporting affine functionals for Entanglement of Formation

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Aug 27, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-co-developed. Editorial metadata: CC BY 4.0; linked content retains its own rights.

Canonical aliases: Supporting affine functionals for Entanglement of Formation · Entanglement of Formation supporting functional

Confidence: Not scored

Registry verification: site confirmed · preprint · resolved

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VibeMathed
registry · event recorded by

Claude Fable 5
model · ai model contributor · Anthropic

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VibeMathed record: Supporting affine functionals for Entanglement of Formation

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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 $\rho_t$ . Results. $C(\rho)=1/2$ 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. What 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.

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Supporting affine functionals for Entanglement of Formation

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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This event attributed to Claude Fable 5

Supporting affine functionals for Entanglement of Formation parent of this event

For two qubits, the paper considers $\rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01|$ , where $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ . This rank-2 state has no global supporting affine functional for Entanglement of Formation. Setting $\rho_t=(1-t)\rho+t|10\rangle\langle10|$ , Wootters’ formula gives $C(\rho_t)=\frac12-\sqrt{2t}+O(t)$ . Consequently, $\lim_{t\to0^+}[E_F(\rho)-E_F(\rho_t)]/t=+\infty$ , so $E_F$ 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. parent of this event

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