On the coincidence of discrete and continuous definitions of Entanglement of Formation

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the identification of a Borel ensemble of pure bipartite states with a Borel probability measure on the unit sphere of the space of Hilbert–Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.