VibeMathedMath problems solved with AI
Sign in

Supporting affine functionals for Entanglement of Formation

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 ρ\rho for which no Hermitian Λρ\Lambda_\rho satisfies both EF(ρ)=TrΛρρE_F(\rho)=\mathrm{Tr}\Lambda_\rho\rho and EF(σ)≥TrΛρσE_F(\sigma)\geq\mathrm{Tr}\Lambda_\rho\sigma for every state σ\sigma. The construction uses the equivalence between existence of such a functional and Lipschitz lower semicontinuity of EFE_F, together with Wootters’ formula.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Construction
Field
Entanglement theory
Posed by
A.S. Holevo and M.E. Shirokov
Year posed
—
Years open
—
Solved
2026-08-27
Model
Claude Fable 5
Vendor
Anthropic
Collaborators
—
Verification
Site-confirmed
Publication
Preprint
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For two qubits, the paper considers

ρ=12∣Φ+⟩⟨Φ+∣+12∣01⟩⟨01∣\rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01|,

where ∣Φ+⟩=(∣00⟩+∣11⟩)/2|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2. This rank-2 state has no global supporting affine functional for Entanglement of Formation.

Setting ρt=(1−t)ρ+t∣10⟩⟨10∣\rho_t=(1-t)\rho+t|10\rangle\langle10|, Wootters’ formula gives

C(ρt)=12−2t+O(t)C(\rho_t)=\frac12-\sqrt{2t}+O(t).

Consequently,

lim⁡t→0+[EF(ρ)−EF(ρt)]/t=+∞\lim_{t\to0^+}[E_F(\rho)-E_F(\rho_t)]/t=+\infty,

so EFE_F is not Lipschitz lower semicontinuous at ρ\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.

What the AI did

From the abstract: "We use Wootters' formula and the help of Claude Fable 5 to find a state ρ\rho of the system ABAB 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.

AI-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'.

Verification

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 ρ=12∣Φ+⟩⟨Φ+∣+12∣01⟩⟨01∣\rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01| and on ρt\rho_t.

Results. C(ρ)=1/2C(\rho)=1/2 exactly. C(ρt)C(\rho_t) has exact closed form at each rational tt - at t=10−6t=10^{-6} it is 999999/2000000−3222222/106999999/2000000-3\sqrt{222222}/10^6, which differs from 12−2t\frac12-\sqrt{2t} by −4.99×10−7-4.99\times10^{-7}, the O(t)O(t) term with coefficient about −12-\frac12, confirming the paper's expansion. The difference quotient [EF(ρ)−EF(ρt)]/t[E_F(\rho)-E_F(\rho_t)]/t evaluates to 14.8114.81, 154.7154.7, 1550.91550.9, 15512.715512.7 and 155131.3155131.3 at t=10−2,10−4,10−6,10−8,10−10t=10^{-2},10^{-4},10^{-6},10^{-8},10^{-10}: a factor of ten per two decades, so it grows like t−1/2t^{-1/2} and diverges. EFE_F is therefore not Lipschitz lower semicontinuous at ρ\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.

Source

Submitted by VibeGene on

Changelog2 changes

Discussion