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arXiv:2610.35797

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

M. E. Shirokov

quant-phmath-phmath.FASubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations
v1 · Oct 26, 2026v2 · Oct 30, 2026
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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.

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36 pages, preliminary version, one comments and references are welcome

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The ε\varepsilonε-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation
We describe a family of convex continuous functions EF,εE_{F,\varepsilon}EF,ε​, ε∈(0,1]\varepsilon\in(0,1]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\varepsilonε-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\varepsilonε and converge pointwise to the EoF as ε→0+\varepsilon\to0^+ε→0+. We conducted a detailed analysis of properties of the functions EF,εE_{F,\varepsilon}EF,ε​, using (among others) the relationship between the family {EF,ε}ε∈[0,1]\{E_{F,\varepsilon}\}_{\varepsilon\in[0,1]}{EF,ε​}ε∈[0,1]​ and the family {EFλ(ρ)}λ>0\{E^λ_F(ρ)\}_{λ>0}{EFλ​(ρ)}λ>0​ of Moreau-Yosida approximations of the EoF explored in [arXiv:2609.30246].
We obtain a variational expression for the functions EF,εE_{F,\varepsilon}EF,ε​ (which for ε=0\varepsilon=0ε=0 turns into the standard variational expression for the EoF). We also describe the weak subadditivity property of the functions EF,εE_{F,\varepsilon}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,εE_{F,\varepsilon}EF,ε​ to the EoF as ε→0+\varepsilon\to0^+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.
We describe a family of convex uniformly continuous functions EFλE^λ_FEFλ​, λ>0λ>0λ>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→0λ\to0λ→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 EFλE^λ_FEFλ​ 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)−EFλ(ρ)E_F(ρ)-E^λ_F(ρ)EF​(ρ)−EFλ​(ρ) for a given state ρρρ. These bounds give easily computable bounds on the rate of uniform convergence of the function EFλE^λ_FEFλ​ to the EoF as λ→0+λ\to0^+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)E_F(ρ)EF​(ρ) and EFλ(ρ)E^λ_F(ρ)EFλ​(ρ) at a given state ρρρ for all λλλ small enough and consider several classes of states for which such coincidence takes place. The conjectured selective LOCC-monotonicity of the functions EFλE^λ_FEFλ​ and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.
In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems AAA and BBB guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system ABABAB. This means that for any state ρρρ of ABABAB there is a Hermitian operator ΛρΛ_ρΛρ​ on HAB=HA⊗HB\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_BHAB​=HA​⊗HB​ such that EF(ρ)=TrΛρρE_F(ρ)=\mathrm{Tr}Λ_ρρEF​(ρ)=TrΛρ​ρ and EF(σ)≥TrΛρσE_F(σ)\geq\mathrm{Tr}Λ_ρσEF​(σ)≥TrΛρ​σ for any state σσσ of ABABAB. We present an explicit example showing that, when ρρρ is degenerate, this is not true even in the simplest case when AAA and BBB are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state ρρρ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state ρρρ. We use Wootters' formula and the help of Claude Fable 5 to find a state ρρρ of the system ABABAB for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state ρρρ (i.e. inequalities of the form  EF(ρ)−EF(σ)≤Cρ∥ρ−σ∥1\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1EF​(ρ)−EF​(σ)≤Cρ​∥ρ−σ∥1​) with and without restrictions on the support of the state σσσ.
In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite states. These characterizations are variational in nature, meaning that separability and EOF are given in terms of a function that is to be minimized over the manifold of unitary matrices. A major benefit of such a characterization is that it directly leads to a numerical procedure for calculating EOF. We present an efficient minimization algorithm and apply it to the bound entangled $3\ifmmode\times\else\texttimes\fi{}3$ Horodecki states; we show that their EOF is very low and that their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)] on EOF for $2\ifmmode\times\else\texttimes\fi{}2$ states and also present some progress in generalizing these results to higher-dimensional systems.
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρρρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality  S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) \,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,S(ρ)−S(σ)≤Cρ​(1−F(ρ,σ)) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the rank of ρρρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρρρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality  EF(ρ)−EF(σ)≤Cρ(1−Trρσ) \,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,EF​(ρ)−EF​(σ)≤Cρ​(1−Trρσ) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the Schmidt rank of ρρρ. In both cases the optimal constant CρC_ρCρ​ is equal to the optimal constant KdK_{d}Kd​ in the log-Sobolev inequality for the complete graph with ddd vertices: in the first case d=rankρd=\mathrm{rank}ρd=rankρ, in the second one d=rankρA=rankρBd=\mathrm{rank}ρ_A=\mathrm{rank}ρ_Bd=rankρA​=rankρB​. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
We analyse two possible definitions of the squashed entanglement in an infinite-dimensional bipartite system: direct translation of the finite-dimensional definition and its universal extension. It is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy. It is also shown that the second definition gives an adequate lower semicontinuous extension of this measure to all states of the infinite-dimensional bipartite system. A general condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is proved and its corollaries are considered. Continuity bound for the squashed entanglement under the energy constraint on one subsystem is obtained by using the tight continuity bound for quantum conditional mutual information (proved in the Appendix by using Winter’s technique). It is shown that the same continuity bound is valid for the entanglement of formation. As a result the asymptotic continuity of the both entanglement measures under the energy constraint on one subsystem is proved.
We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still trace-norm dense in state space, even under the constraint of a finite mean energy.