The theorem also includes a converse: if two quantum states do commute, there is a method for broadcasting them: they must have a common basis of eigenstates diagonalizing them simultaneously, and the map that clones every state of this basis is a legitimate quantum operation, requiring only physical resources independent of the input state to implement—a completely positive map.
The theorem also states that if two quantum states commute, there's a method to broadcast them: they must share a common basis of eigenstates that diagonalize them simultaneously, and the mapping that duplicates every state in this basis is a legitimate quantum operation that can be implemented using only physical resources independent of the input state.