LOCC-average monotonicity of Hilbert–Schmidt entanglement

Establish whether the Hilbert–Schmidt entanglement E_HS(ρ)=min_{σ∈S_sep}∥ρ−σ∥²₂ is nonincreasing on average under LOCC, or provide a counterexample.

Background

E_HS is defined by the squared Hilbert–Schmidt distance to the separable set. It was introduced as a candidate measure, but its LOCC-average monotonicity is unresolved.

The authors note numerical investigations found no counterexamples but the property remains unknown.