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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 butnot a sufficient condition for positivity of $\Phi_1 \otimes \Phi_2$. We find anon-trivial sufficient condition for positivity of the tensor product mapbeyond the cases when both $\Phi_1$ and $\Phi_2$ are completely positive orcompletely co-positive. We find necessary and (separately) sufficientconditions for $n$-tensor-stable positive qubit maps, i.e. such qubit maps$\Phi$ that $\Phi^{\otimes n}$ is positive. Particular cases of 2- and3-tensor-stable positive qubit maps are fully characterized, and thedecomposability of 2-tensor-stable positive qubit maps is discussed. The caseof non-unital maps is reduced to the case of appropriate unital maps. Finally,$n$-tensor-stable positive maps are used in characterization of multipartiteentanglement, namely, in the entanglement depth detection.

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