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
Λρ on
HAB=HA⊗HB such that
EF(ρ)=TrΛρρ and
EF(σ)≥TrΛρσ 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
EF(ρ)−EF(σ)≤Cρ∥ρ−σ∥1) with and without restrictions on the support of the state
σ.
An important measure of bipartite entanglement is the entanglement of formation, which is defined as the minimum average pure state entanglement of all decompositions realizing a given state. A decomposition which achieves this minimum is called an optimal decomposition. However, as for the entanglement of formation, there is not much known about the structure of such optimal decompositions, except for some special cases, like states of two qubits or isotropic states. Here we present a necessary and sufficient condition for a set of pure states of a finite dimensional bipartite system to form an optimal decomposition. This condition is well suited to treat the question, whether the entanglement of formation is additive or not.
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 a new method of constructing sets of states with known entanglement of formation. The method realizes the optimal decomposition families of states. We also derive the additivity of the entanglement of formation and thus calculate the entanglement cost. These results are used to find out the undistillable entanglement of some states. We illustrate our method by investigating the two-qubit state, the separable state, the maximally correlated state, the isotropic state and the Werner state. We also discuss the realization of Werner state in experiment.