Researchers have demonstrated that a widely accepted assumption in quantum information theory, concerning the existence of supporting affine functionals for the Entanglement of Formation (EoF), is not always valid. This assumption stated that, for any quantum state of a bipartite system, a global supporting affine functional always exists. However, the new work presents an explicit counterexample that invalidates this belief, even for the simplest case of two entangled qubits.

The Entanglement of Formation (EoF) is a crucial measure of quantum entanglement, quantifying the minimum amount of entanglement needed to prepare a given state. The existence of a global supporting affine functional would imply that the EoF behaves 'smoothly' across the entire state space, facilitating its analysis and calculation. The refutation of this assumption has significant implications for the theoretical understanding of entanglement and for the development of methods for its quantification.

The counterexample is based on the equivalence between the existence of a supporting affine functional and the Lipschitz lower semicontinuity of the EoF at a given state. Using Wootters' formula, the authors constructed a degenerate state for which this semicontinuity property does not hold. This finding demonstrates that the convex roof structure of the EoF and the finite dimensionality of the subsystems do not, by themselves, guarantee the existence of such functionals. The study also describes the conditions under which local and global supporting affine functionals do exist for both finite and infinite-dimensional bipartite quantum systems, and establishes Lipschitz lower semicontinuity bounds for finite-rank states.