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

Abstract

We analyze positivity of a tensor product of two linear qubit maps, . Positivity of maps and is a necessary but not a sufficient condition for positivity of . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both and 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 that 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.