  We analyze positivity of a tensor product of two linear qubit maps, $\Phi_1
\otimes \Phi_2$. Positivity of maps $\Phi_1$ and $\Phi_2$ is a necessary but
not a sufficient condition for positivity of $\Phi_1 \otimes \Phi_2$. We find a
non-trivial sufficient condition for positivity of the tensor product map
beyond the cases when both $\Phi_1$ and $\Phi_2$ are completely positive or
completely co-positive. We find necessary and (separately) sufficient
conditions for $n$-tensor-stable positive qubit maps, i.e. such qubit maps
$\Phi$ that $\Phi^{\otimes n}$ is positive. Particular cases of 2- and
3-tensor-stable positive qubit maps are fully characterized, and the
decomposability of 2-tensor-stable positive qubit maps is discussed. The case
of non-unital maps is reduced to the case of appropriate unital maps. Finally,
$n$-tensor-stable positive maps are used in characterization of multipartite
entanglement, namely, in the entanglement depth detection.
