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

Abstract:We describe a family of convex uniformly continuous functions $E^\lambda_F$, $\lambda>0$, 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 $\lambda\to0$. 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 $E^\lambda_F$ the Moreau-Yosida approximations of the EoF and describe their equivalent definitions and basic properties. The semicontinuity bounds for the EoF (presented in [this http URL., 46(6), 2632-2658]) allow us to obtain easily computable upper bounds on the difference $E_F(\rho)-E^\lambda_F(\rho)$ for a given state $\rho$. These bounds give easily computable bounds on the rate of uniform convergence of the function $E^\lambda_F$ to the EoF as $\lambda\to0^+$ on the sets of states with bounded rank/energy of one of the marginal states. We also discuss sufficient conditions for the coincidence of $E_F(\rho)$ and $E^\lambda_F(\rho)$ at a given state $\rho$ for all $\lambda$ small enough and consider several classes of states for which such coincidence takes place. The conjectured selective LOCC-monotonicity of the functions $E^\lambda_F$ and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.

arXiv:2609.30246 [quant-ph]

(or arXiv:2609.30246v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.30246 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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