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arXiv:2608.27363

On supporting affine functionals for Entanglement of Formation

A. S. Holevo, M. E. Shirokov

quant-phmath-phmath.OCSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $ρ$ of $AB$ there is a Hermitian operator $Λ_ρ$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(ρ)=\mathrm{Tr}Λ_ρρ$ and $E_F(σ)\geq\mathrm{Tr}Λ_ρσ$ for any state $σ$ of $AB$. We present an explicit example showing that, when $ρ$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $ρ$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $ρ$. We use Wootters' formula and the help of Claude Fable 5 to find a state $ρ$ of the system $AB$ 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 $ρ$ (i.e. inequalities of the form $\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1$) with and without restrictions on the support of the state $σ$.

Comments
14 pages, preliminary version, any comments are welcome

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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.
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The ε\varepsilonε-smoothed Entanglement of Formation and its relation to the Moreau-Yosida approximation
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We study the entanglement-breaking (EB) space, such that the entanglement of formation (EOF) of a bipartite quantum state is additive when its range is an EB subspace. We systematically construct the EB spaces in the Hilbert space ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, and the two-dimensional EB space in ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We characterize the expression of two-qubit states of rank two with nonadditive EOF, if they exist. We further apply our results to construct EB spaces of arbitrarily given dimensions. We show that the example in Phys. Rev. Lett. 89, 027901 (2002) is a special case of our results. We further work out the entanglement cost of a qubit-qutrit state in terms of the two-atom system of the Tavis-Cummings model.
We study the entanglement R\'enyi $\ensuremath{\alpha}$ entropy $(\mathrm{ER}\ensuremath{\alpha}\mathrm{E})$ as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entanglement of formation (EoF), the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ gives a continuous spectrum parametrized by variable $\ensuremath{\alpha}$ as the entanglement measure, and it reduces to the standard EoF in the special case $\ensuremath{\alpha}\ensuremath{\rightarrow}1$. The $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ provides more information in entanglement quantification and can be used, for example, to determine the convertibility of entangled states by local operations and classical communication. A series of results is obtained: (i) we show that the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for the EoF of two states; (ii) similar to the case of EoF, we study in a fully analytical way the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ for arbitrary two-qubit states, the Werner states, and isotropic states in the general ddd dimension; and (iii) we provide a proof of the previous conjecture for the analytical functional form of the EoF of isotropic states in the arbitrary ddd dimension.
We give a criterion for the existence of a global supporting affine functional for the entanglement of formation at an entangled two-qubit state. It exists if and only if the spin-flip bilinear form is nondegenerate on the support of the state, or equivalently, if the last concurrence root associated with the support is nonzero. In this case the entanglement of formation satisfies a two-sided Lipschitz bound relative to the reference state. In the degenerate case the state has rank two or three, and its projective support is tangent to the product-state quadric. By suitably choosing a perturbation outside the original support space, the decrease in the entanglement of formation is, to leading order, proportional to the square root of the mixing parameter. We give an explicit expression for the leading coefficient. Such a square-root decrease violates the one-sided Lipschitz bound, and therefore no globally supported affine functional exists at this state.