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

TL;DR

This paper investigates conditions under which tensor products of qubit maps remain positive, providing new criteria for n-tensor-stable positive maps and applications in entanglement detection.

Contribution

It introduces non-trivial sufficient conditions for positivity of tensor products of qubit maps beyond known cases and characterizes n-tensor-stable positive qubit maps for specific n.

Findings

Identified non-trivial sufficient conditions for positivity of tensor products of qubit maps.

Fully characterized 2- and 3-tensor-stable positive qubit maps.

Applied n-tensor-stable positive maps to multipartite entanglement detection.

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.