Positive tensor products of maps and n -tensor-stable positive qubit maps
Sergey N. Filippov, Kamil Yu. Magadov · Journal of Physics A Mathematical and Theoretical · 2017
Abstract We analyze positivity of a tensor product of two linear qubit maps, Φ 1 ⊗ Φ 2 . Positivity of maps Φ 1 and Φ 2 is a necessary but not a sufficient condition for positivity of Φ 1 ⊗ Φ 2 . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ 1 and Φ 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 Φ that Φ ⊗ 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.