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arXiv:2609.30246

The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates

M. E. Shirokov

quant-phmath-phSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations• Revised Sep 30, 2026 (v2)
v1 · Sep 24, 2026v2 · Sep 30, 2026
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Abstract

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 $λ\to0$. 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 $λ\to0^+$ 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.

Comments
25 pages, 2 figures, preliminary version, any comments are welcome

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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.
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.
Special approximation technique for analysis of different characteristics of states of multipartite infinite-dimensional quantum systems is proposed and applied to study of the relative entropy of entanglement and its regularisation. We prove several results about analytical properties of the multipartite relative entropy of entanglement and its regularization (the lower semicontinuity on wide class of states, the uniform continuity under the energy constraints, etc.). We establish a finite-dimensional approximation property for the relative entropy of entanglement and its regularization that allows to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
An universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results concerning the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of πππ-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyse continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
A universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results on the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of π-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyze continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
The entanglement of formation (EOF), which quantifies the required minimally physical resources to prepare a quantum state, plays important roles in many quantum information processes. Due to the convex-roof definition, it is hard to find the explicit analytic formulae of EOF, except for some special quantum systems. The correlation matrix of a quantum state is closely related to entanglement property. In this paper, we use the Lagrange-multiplier approach to obtain an analytic lower bound of EOF for arbitrary dimensional bipartite states, which improves the previous conclusions. Numerical example shows that the bound is also valid for some PPT entangled states.