Bounds on action of local quantum channels

We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on multi-partite quantum states. We assume that the individual channels $\Lambda_j$ affect single-particle states so, that for an arbitrary input $\rho_j$, the distance $D_j (\Lambda_j [ \rho_j ], \rho_j)$ between the input $\rho_j$ and the output $\Lambda_j [ \rho_j ]$ of the channel is less than $\epsilon$. Given this assumption we show that for an arbitrary {\em separable} two-partite state $\rho_{12}$ the distance between the input $\rho_{12}$ and the output $\Lambda_1\otimes\Lambda_2[\rho_{12} ]$ fulfills the bound $D_{12} (\Lambda_1 \otimes \Lambda_2 [ \rho_{12} ], \rho_{12}) \leq \sqrt{2+ 2 \sqrt{(1-1/d_1)(1-1/d_2)}} \epsilon$ where $d_1$ and $d_2$ are dimensions of first and second quantum system respectively. On the contrary, entangled states are transformed in such a way, that the bound on the action of the local channels is $D_{12} (\Lambda_1 \otimes \Lambda_2 [ \rho_{12} ], \rho_{12}) \leq 2 \sqrt{2 - 1/d} \: \epsilon$, where $d$ is the dimension of the smaller of the two quantum systems passing through the channels. Our results show that the fundamental distinction between the set of separable and the set of entangled states results into two different bounds which in turn can be exploited for a discrimination between the two sets of states. We generalize our results to multi-partite channels.