Supporting affine functionals for entanglement of formation

Does entanglement of formation admit a global supporting affine functional at every boundary state? The test state is ρ=(|Φ+⟩⟨Φ+|+|01⟩⟨01|)/2 with |Φ+⟩=(|00⟩+|11⟩)/√2; perturbations toward |10⟩ probe the boundary behavior.

Why it matters and what counts as progress

Why it matters: Failure of a supporting functional exposes boundary behavior that is hidden by assumptions valid for interior entanglement states.

What counts as progress: Construct a state with no supporting functional or characterize the boundary conditions where supporting functionals do exist.

How the problem developed

1998 · Formula [1]
William K. Wootters Derived the closed formula for entanglement of formation of two-qubit states, later used to search for a degenerate state without a global supporting affine functional.
27 Aug 2026 · Counterexample [2]
Alexander S. Holevo and Mikhail E. Shirokov, with Claude Fable 5 assistance Gave an explicit degenerate two-qubit state at which entanglement of formation has no global supporting affine functional, using Wootters’ formula and a failure of local Lipschitz lower semicontinuity.

AI contribution

On supporting affine functionals for Entanglement of Formation

A. S. Holevo, M. E. Shirokov

For the two-qubit state ρ=(|Φ+⟩⟨Φ+|+|01⟩⟨01|)/2, where |Φ+⟩=(|00⟩+|11⟩)/√2, Holevo and Shirokov show that entanglement of formation has no global supporting affine functional. Along a perturbation toward |10⟩, it decreases on a square-root scale.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFull