The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates
Abstract
We describe a family of convex uniformly continuous functions EλF, λ>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→0. These functions are "nonselective" entanglement monotones defined in a 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 EλF 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 EF(ρ)−EλF(ρ) for a given state ρ. These bounds give easily computable bounds on the rate of uniform convergence of the function EλF to the EoF as λ→0+ on the sets of states with bounded rank/energy of one of the marginal states. We also discuss sufficient conditions for the coincidence of EF(ρ) and EλF(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 EλF and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.
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BibTeX
@misc{shirokov2026moreauyosida,
title = {{The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates}},
author = {M. E. Shirokov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30246},
eprint = {2609.30246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30246},
}
RIS
TY - UNPB
TI - The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates
AU - M. E. Shirokov
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.30246
UR - https://arxiv.org/abs/2609.30246
AB - We describe a family of convex uniformly continuous functions E^λ_F, λ>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→0. These functions are "nonselective" entanglement monotones defined in a 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^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^λ_F(ρ) for a given state ρ. These bounds give easily computable bounds on the rate of uniform convergence of the function E^λ_F to the EoF as λ→0^+ 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^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^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.
ER -