Selective LOCC monotonicity of the Moreau-Yosida approximation

Prove or disprove that, for every order parameter \(\lambda>0\), the Moreau-Yosida approximation \(E_F^\lambda\) of the Entanglement of Formation is non-increasing under selective LOCC operations.

Background

The paper proves that EFλE_F^\lambda is convex, Lipschitz continuous, vanishes exactly on separable states, and is non-increasing under nonselective LOCC operations. Selective LOCC monotonicity would establish that EFλE_F^\lambda is a full entanglement monotone in the standard sense.

The authors formulate a conjecture that selective LOCC monotonicity also holds. They reduce a sufficient route to proving it to a flagged-state equality: the minimizing state, or a minimizing sequence, in the definition of EFλE_F^\lambda should preserve the local classical flag and its probability distribution. A general proof or counterexample is not provided.

References

It is proved in Section 3 (Corollary 1) that the function $E{\lambda}_F$ does not increase under nonselective LOCC-operations for every $\lambda>0$. Unfortunately, the (quite simple) arguments used to prove this property are not generalized to selective LOCC-operations. At the same time, all the attempts of ChatGPT-5.6 to find a counterexample were unsuccessful. So, we may conjecture, at the moment, that the function $E{\lambda_F$ does not increase under selective LOCC-operations} as well.

— The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates  (2609.30246 - Shirokov, 24 Sep 2026) in Section 7, “Open question: can the function $E^{\lambda}_F$ increase under selective LOCC-operations?”