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.

A universal framework for entanglement detection under group symmetry

“…These channels are precisely the covariant channels with respect to the fundamental representation of the signed symmetric group (or the hyperoctahedral group) H d (proposition 4.3). Our result strengthens some conclusions in [VW01,KMS20] from the following two perspectives:…”

“…(2) For the proof of theorem 4.1, we show that every positive linear map of the form (1.3) is decomposable (theorem 4.5). This resolves some of the open questions posed in [KMS20]. Specifically, the authors of [KMS20] questioned whether all positive maps of the form…”

“…This resolves some of the open questions posed in [KMS20]. Specifically, the authors of [KMS20] questioned whether all positive maps of the form…”

A class of optimal positive maps in $M_n$

“…maximal commutative subgroup of the special unitary group SU (n). Covariant maps were recently analyzed in [33][34][35].…”

Irreducibly $SU(2)$-covariant quantum channels of low rank

“…However, positivity of Φ is not immediate from this decomposition, and such a question of characterizing positive irreducibly covariant maps was raised as an open problem in [MSD17]. Subsequently, positive irreducibly covariant linear maps for the permutation group S(3) and the quaternion group Q have been characterized in [KMS20].…”