The $\varepsilon$-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation

Abstract

We describe a family of convex continuous functions $E_{F,\varepsilon}$, $\varepsilon\in(0,1]$, 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 $\varepsilon$-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 $\varepsilon$ and converge pointwise to the EoF as $\varepsilon\to0^+$. We conducted a detailed analysis of properties of the functions $E_{F,\varepsilon}$, using (among others) the relationship between the family $\{E_{F,\varepsilon}\}_{\varepsilon\in[0,1]}$ 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 $E_{F,\varepsilon}$ (which for $\varepsilon=0$ turns into the standard variational expression for the EoF). We also describe the weak subadditivity property of the functions $E_{F,\varepsilon}$ 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 $E_{F,\varepsilon}$ to the EoF as $\varepsilon\to0^+$ 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.

The paper studies trace-distance smoothing of entanglement of formation (EoF), $E_{F,\varepsilon}(\rho)=\inf_{T(\rho,\sigma)\leq\varepsilon}E_F(\sigma)$, in finite- and infinite-dimensional bipartite systems, and its relationship to the penalized approximation $E_F^\lambda(\rho)=\inf_\sigma\{E_F(\sigma)+T(\rho,\sigma)/\lambda\}$. Convex duality gives $E_F^\lambda=\min_{0\leq\varepsilon\leq1}(E_{F,\varepsilon}+\varepsilon/\lambda)$ and $E_{F,\varepsilon}=\sup_{\lambda>0}(E_F^\lambda-\varepsilon/\lambda)$, yielding an operator variational representation. For every positive smoothing radius, the smoothed EoF is finite, convex and trace-norm continuous, including in infinite dimensions. It decreases under nonselective LOCC and converges pointwise to EoF as $\varepsilon\downarrow0$. Its zero set consists exactly of states within distance $\varepsilon$ of separability, with $E_{F,\varepsilon}(\rho)\geq2[d(\rho,\mathrm{Sep})-\varepsilon]_+^2$. An auxiliary weak-* compactness argument proves that every infinite-dimensional bipartite state has a nearest separable state in trace distance. Smoothing satisfies $E_{F,\varepsilon_1+\varepsilon_2}(\rho\otimes\sigma)\leq E_{F,\varepsilon_1}(\rho)+E_{F,\varepsilon_2}(\sigma)$, but fixed-radius subadditivity fails. Existing semicontinuity bounds imply uniform convergence under marginal rank or energy constraints; for marginal rank $d\geq2$, the error is at most $\delta\ln(d-1)+h^\uparrow(\delta)$, where $\delta=\sqrt{\varepsilon(2-\varepsilon)}$. Smoothing and regularization do not commute. For a maximally entangled state $\Omega_d$, the regularized smoothed EoF equals $(1-\varepsilon)\ln d$, whereas fixed-error operational preparation costs $\ln d$ for $\varepsilon<1$. Infinite-dimensional smoothing minimizers are not established, selective LOCC monotonicity can fail, and a proposed general ordering between smoothing and regularization remains open. Several structural results extend to relative entropy of entanglement.