On the structure of Completely Reducible States

Abstract: The complete reducibility property for bipartite states reduced the separability problem to a proper subset of positive under partial transpose states and was used to prove several theorems inside and outside entanglement theory. So far only three types of bipartite states were proved to possess this property. In this work, we provide some procedures to create states with this property, we call these states by the name of completely reducible states. The convex combination of such states is the first procedure, showing that the set of completely reducible states is a convex cone. We also provide a complete description of the extreme rays of this set. Then we show that powers, roots and partial traces of completely reducible states result in states of the same type. Finally, we consider a shuffle of states that preserves this property. This shuffle allows us to construct states with the complete reducibility property avoiding the only three conditions known to date that imply this property.