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

Abstract

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.

11 pages, 7 figures. Section on entanglement witnessing is added

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• Decomposability of Linear Maps under Tensor Products
• Lower and upper bounds on nonunital qubit channel capacities
• Halos and undecidability of tensor stable positive maps
• Spin polarization-scaling quantum maps and channels
• Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps
• Superactivation of memory effects in a classical Markov environment