bibkey: shirokov2026moreauyosidaeof authors: M. E. Shirokov year: 2026 title: “The Moreau-Yosida approximation of the EoF: basic properties and accuracy estimates” doi: null url: https://arxiv.org/abs/2609.30246v1 claim: “we may conjecture, at the moment, that the function E_F^lambda does not increase under selective LOCC-operations” strata_touched:

• D5/S3/Quantum/Entanglement/MoreauYosidaFormationSelectiveLoccRefutation license: citation-only triage: anchor

The Moreau-Yosida approximation of the EoF

M. E. Shirokov, arXiv:2609.30246v1, Section 7, p. 20.

Verified locator

https://arxiv.org/abs/2609.30246v1

Statement

The finite-dimensional entanglement of formation is defined in (EF-d), equation (1), p. 2, by

$$E_F(\rho )=\underset{{\sum }_k{p}_k{\varrho }_k=\rho }{\inf }\sum _k{p}_{k}S([{\varrho }_k{]}_A).$$

where the infimum is taken over all finite ensembles $\{p_k,{\varrho }_k\}$ of pure states in $\mathfrak{S}({\mathcal{H}}_{AB})$ having $\rho$ as their average state

The entropy uses the natural logarithm. Equation (EFA), equation (13), p. 6, defines, for $\lambda >0$,

$$E_F^{\lambda }(\rho )=\underset{\sigma \in \mathfrak{S}({\mathcal{H}}_{AB})}{\inf }\left\{E_F(\sigma )+\frac{1}{2\lambda }\Vert \rho -\sigma {\Vert }_1\right\}.$$

The source footnote on p. 7 specifies:

It is essential that we use $\Vert \rho -\sigma {\Vert }_1$ instead of $\Vert \rho -\sigma {\Vert }_1^2$

Section 7 asks:

Open question: can the function $E_F^{\lambda }$ increase under selective LOCC-operations?

The conjectural sentence is:

It is proved in Section 3 (Corollary 1) that the function $E_F^{\lambda }$ 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 $E_F^{\lambda }$ does not increase under selective LOCC-operations as well.

The formal claim uses the standard selective inequality: the probability-weighted sum of the normalized branch values is at most the input value. It quantifies over finite dimensions and finite-round local Kraus instruments, with several Kraus operators allowed per classical outcome and the full classical outcome history retained. The one-round two-outcome Alice projective measurement is a member of this class, and also a member of the larger LOCC classes used in the source.

The formal refutation gives an input on ${\mathbb{C}}^5\otimes {\mathbb{C}}^3$ and $\lambda =7/2$ with input value at most $1/7$ and output average at least $3/20$. It uses the full finite pure-state convex roof and the unsquared trace norm. The pure-state cost is literally the von Neumann entropy of Alice’s reduced density state, with the frozen rank-one density-state constructor and partial trace. Both formation functions take values in $[0,+\infty ]$ via ENNReal.ofReal of their nonnegative real expressions for $\lambda >0$.