A theoretical work describes a family of convex and continuous functions, denoted E_F,ε, defined on the states of a bipartite quantum system with finite- or infinite-dimensional subsystems. Built using the ε-smoothed measure construction proposed by Mora, Piani and Briegel (Phys. Rev. A 109, 052413) applied to the Entanglement of Formation (EoF) with the trace norm as the underlying metric, these functions are non-increasing in ε and converge pointwise to the EoF as ε tends to 0⁺.

The analysis explores the relationship between the family {E_F,ε} and the Moreau-Yosida approximations E_F^λ(ρ) of the Entanglement of Formation, previously studied by the authors. As a central result, the work derives a variational expression for E_F,ε which, in the limit ε = 0, reduces to the standard variational expression for the EoF. The paper also establishes a weak subadditivity property and examines in detail the continuity properties of these functions.

Exploiting semicontinuity bounds for the EoF published in Lobachevskii Journal of Mathematics (46, 2632-2658) and in arXiv:2404.07978, the authors derive easily computable upper bounds on the rate of uniform convergence of E_F,ε to the EoF as ε tends to 0⁺. These bounds apply to sets of states whose marginal has bounded rank or energy, a scenario relevant for infinite-dimensional quantum systems.

The work also discusses the extension of these results to other entanglement measures, in particular to the relative entropy of entanglement. As an auxiliary result, the authors prove that for any state of an infinite-dimensional bipartite quantum system there always exists a nearest separable state with respect to the trace norm distance, although such a state need not be unique. The existence and non-uniqueness of this separable projector provides useful geometric information for characterising the set of separable states in infinite dimensions.