Positive tensor products of qubit maps and n-tensor-stable positive qubit maps

We analyze positivity of a tensor product of two linear qubit maps, $\Phi_1 \otimes \Phi_2$\mathrm{.}Positivityofmaps$\Phi_1$and$\Phi_2$ is a necessary but not a sufficient condition for positivity of ${\mathrm{\Phi }}_1\otimes {\mathrm{\Phi }}_2$. We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ 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 $\mathrm{\Phi }$ that ${\mathrm{\Phi }}^{\otimes n}$ 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.