Positive Maps From Irreducibly Covariant Operators

Abstract

In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different kind. In particular, a family of necessary conditions for positivity, for such objects is derived, stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map. In the low-dimensional cases, for the permutation group $S(3)$ and the quaternion group $Q$, the necessary and sufficient conditions are given, together with the discussion on their decomposability. In higher dimensions, we present positive maps induced by a three-dimensional irreducible representation of the permutation group $S(4)$ and $d$-dimensional representation of the monomial unitary group $MU(d)$. In the latter case, we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction. We show that the generalised Choi map can be obtained by considered covariant maps induced by the monomial unitary group. As an additional result, a novel interpretation of the Fujiwara-Algolet conditions for positivity and complete positivity is presented. Finally, in the end, a new form of an irreducible representation of the symmetric group $S(n)$ is constructed, allowing us to simplify the form of certain Choi-Jamiołkowski images derived for irreducible representations of the symmetric group.