  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.
