Small-parameter equality for all pure states of a two-qubit system

Determine whether, for every pure state of the simplest two-qubit system, there exists a positive \(\lambda_\rho\) such that \(E_F^\lambda(\rho)=E_F(\rho)\) for all \(0<\lambda\leq\lambda_\rho\).

Background

The paper characterizes equality EFλ(ρ)=EF(ρ)E_F^\lambda(\rho)=E_F(\rho) for sufficiently small λ\lambda through a global affine supporting functional or, equivalently, a suitable semicontinuity bound at the state. It gives sufficient conditions for several classes of states, including certain maximally entangled pure states.

However, even in the two-qubit case, the paper does not determine whether every pure state has such a positive threshold. Thus, the existence of a positive threshold remains unresolved for the full class of pure two-qubit states.

References

It is somewhat surprising that, at the moment, we cannot prove or disprove the validity of (\ref{equ}) with some $\,\lambda_\rho>0\,$ even for all pure states of the simplest 2-qubit system (see the end of the Introduction in ).

— The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates  (2609.30246 - Shirokov, 24 Sep 2026) in Section 6, “The cases when $E_F(\rho)=E^\lambda_F(\rho)$ for small $\lambda>0$,” Case A