The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates

Abstract: We describe a family of convex uniformly continuous functions $E<sup>λ_F$, $λ&gt;0$, on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems), which monotonically increase and converge pointwise to the Entanglement of Formation (EoF) as $λ\to0$. These functions are "nonselective" entanglement monotones defined by the way close to the construction of the Moreau-Yosida regularization (the Moreau envelope) of a convex function on a convex set used in the modern convex analysis. So, we call the functions $E<sup>λ_F$ the Moreau-Yosida approximations of the EoF and describe their equivalent definitions and basic properties. The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46(6), 2632-2658]) allow us to obtain easily computable upper bounds on the difference $E_F(ρ)-E<sup>λ_F(ρ)$ for a given state $ρ$. These bounds give easily computable bounds on the rate of uniform convergence of the function $E<sup>λ_F$ to the EoF as $λ\to0<sup>+$ on the sets of states with bounded rank/energy of one of the marginal states. We also discuss sufficient conditions for the coincidence of $E_F(ρ)$ and $E<sup>λ_F(ρ)$ at a given state $ρ$ for all $λ$ small enough and consider several classes of states for which such coincidence takes place. The conjectured selective LOCC-monotonicity of the functions $E<sup>λ_F$ and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.