The ε-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation
Abstract
We describe a family of convex continuous functions EF,ε, ε∈(0,1], on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems) obtained by applying the Mora-Piani-Briegel construction of ε-measure (proposed in [Phys. Rev. A 109, 052413]) to the Entanglement of Formation (EoF) with the use of the trace norm distance in the role of basic metric. These functions are non-increasing in ε and converge pointwise to the EoF as ε→0+. We conducted a detailed analysis of properties of the functions EF,ε, using (among others) the relationship between the family {EF,ε}ε∈[0,1] and the family {EλF(ρ)}λ>0 of Moreau-Yosida approximations of the EoF explored in [arXiv:2609.30246]. We obtain a variational expression for the functions EF,ε (which for ε=0 turns into the standard variational expression for the EoF). We also describe the weak subadditivity property of the functions EF,ε and analyse their continuity properties. The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46, 2632-2658],[arXiv:2404.07978]) allow us to obtain easily computable upper bounds on the rate of uniform convergence of the function EF,ε to the EoF as ε→0+ on the sets of states with bounded rank/energy of one of the marginal states. Applicability of the obtained results to other entanglement measures, in particular, to the relative entropy of entanglement, is discussed. As an auxiliary result, it is shown that for any state of an infinite-dimensional bipartite quantum system, there exists a nearest separable state w.r.t. the trace norm distance. The non-uniqueness of such a state is also shown.
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BibTeX
@misc{shirokov2026εsmoothed,
title = {{The $\varepsilon$-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation}},
author = {M. E. Shirokov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35797},
eprint = {2610.35797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35797},
}
RIS
TY - UNPB
TI - The ε-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation
AU - M. E. Shirokov
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35797
UR - https://arxiv.org/abs/2610.35797
AB - We describe a family of convex continuous functions E_F,ε, ε∈(0,1], on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems) obtained by applying the Mora-Piani-Briegel construction of ε-measure (proposed in [Phys. Rev. A 109, 052413]) to the Entanglement of Formation (EoF) with the use of the trace norm distance in the role of basic metric. These functions are non-increasing in ε and converge pointwise to the EoF as ε→0^+. We conducted a detailed analysis of properties of the functions E_F,ε, using (among others) the relationship between the family {E_F,ε}_ε∈[0,1] and the family {E^λ_F(ρ)}_λ>0 of Moreau-Yosida approximations of the EoF explored in [arXiv:2609.30246]. We obtain a variational expression for the functions E_F,ε (which for ε=0 turns into the standard variational expression for the EoF). We also describe the weak subadditivity property of the functions E_F,ε and analyse their continuity properties. The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46, 2632-2658],[arXiv:2404.07978]) allow us to obtain easily computable upper bounds on the rate of uniform convergence of the function E_F,ε to the EoF as ε→0^+ on the sets of states with bounded rank/energy of one of the marginal states. Applicability of the obtained results to other entanglement measures, in particular, to the relative entropy of entanglement, is discussed. As an auxiliary result, it is shown that for any state of an infinite-dimensional bipartite quantum system, there exists a nearest separable state w.r.t. the trace norm distance. The non-uniqueness of such a state is also shown.
ER -