Supporting affine functionals for entanglement of formation need not exist
An explicit two-qubit state has no global supporting affine functional for the entanglement of formation, contradicting an assumption used in several papers.
- Lab
- Independent
- Model
- Claude Fable 5
- Field
- Physics
- Date
- 2026-08-27
- Human collaborators
- A. S. Holevo, M. E. Shirokov
Sources
Original work
Independent commentary
What was found
Several articles assume that the convex roof structure of the entanglement of formation together with finite-dimensional subsystems guarantees a global supporting affine functional at every state, meaning a Hermitian operator whose trace against the state gives the entanglement of formation and which lower-bounds it everywhere else. The paper gives an explicit counterexample at a degenerate state, in the simplest case where both subsystems are qubits. The construction rests on the equivalence between existence of such a supporting functional at a state and Lipschitz lower semicontinuity of the entanglement of formation there, and uses Wootters' closed formula to locate the state.
Novelty check
The target is not a named conjecture but an assumption in circulation, which the paper identifies as appearing in several articles by the authors themselves among others, so the thing being corrected is stated before it is refuted. The counterexample is at the smallest possible size, two qubits, where Wootters' formula gives the entanglement of formation in closed form and the claim can therefore be checked directly rather than argued asymptotically. No prior counterexample appears.
Caveats and known objections
A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-assisted, the weakest reading the disclosures support, because every mention of the model in the paper is of assistance in a search step inside a human-built argument: the authors use Wootters' formula and the help of Claude Fable 5 to find a state where the property fails, note that this is done very quickly with the model, and attribute one crucial corollary to work obtained with its help. The equivalence with Lipschitz lower semicontinuity, which is what makes the search possible, is the authors' own.
Also recorded at
vibemathedsupporting-affine-functionals-for-entanglement-of-formation
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
Flag this for triage
Signals order the review queue and nothing else. They are never published, and they never move a grade: that takes a citation.
Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded author verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Supporting affine functionals for entanglement of formation need not exist. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-27-entanglement-supporting-functionalBibTeX
@misc{whataifound-independent-2026-functional,
title = {Supporting affine functionals for entanglement of formation need not exist},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-27-entanglement-supporting-functional}
}