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 $\rho$ 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.

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.