On supporting affine functionals for Entanglement of Formation

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 $\rho$ of $AB$ there is a Hermitian operator $\Lambda_\rho$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(\rho)=\mathrm{Tr}\Lambda_\rho\rho$ and $E_F(\sigma)\geq\mathrm{Tr}\Lambda_\rho\sigma$ for any state $\sigma$ of $AB$. We present an explicit example showing that, when $\rho$ 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 $\rho$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $\rho$. We use Wootters' formula and the help of Claude Fable 5 to find a state $\rho$ 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 $\rho$ (i.e. inequalities of the form $\,E_F(\rho)-E_F(\sigma)\leq C_\rho\|\rho-\sigma\|_1$) with and without restrictions on the support of the state $\sigma$.