exists such that = AB C , \[phipsi\] where operator $A$ contains contributions coming from any round in which party A acted on its subsystem, and similarly for operators $B$ and $C$ [^3]. Carrying on with the 3-qubit example, let us now consider for simplicity that both states ${\mbox{$| \psi \rangle$}}$ and ${\mbox{$| \phi \rangle$}}$ have rank 2 reduced density matrices $\rho_A \equiv $ tr$_{BC}(|\psi\rangle\langle\psi|), \rho_B$ and $\rho_C$. Then clearly the rank of operators $A$, $B$ and $C$ need to be 2 (see appendix A). In other words, each of these operators is necessarily invertible, and in particular = A\^[-1]{}B\^[-1]{}C\^[-1]{} . We see thus that, under the assumption of maximal rank for the reduced density matrices, two-way convertibility implies the existence of invertible operators $A$, $B$ and $C$ as in (\[phipsi\]) \[actually, one-way convertibility alone has already implied that an invertible local operator (ILO) $A\otimes B\otimes C$ exists\]. Obviously, the converse also holds, namely that if an ILO $A\otimes B\otimes C$ exists then for each direction of the conversion a local protocol can be build that succeeds with non-zero probability. As explained in appendix A in detail, we can get rid of the previous assumption on the ranks and announce 