Generalized Choi-Kraus dilations of linear maps between matrix algebras

Łukasz Skowronek, Erling Størmer

Xin Li, Wei Wu

Seung-Hyeok Kye

Satvik Singh, Ion Nechita

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

Priyabrata Bag, Azad Rohilla, Harsh K. Trivedi

If the symmetry, (an operator $J$ satisfying $J=J^*=J^{-1}$) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an $S$-space which was introduced by Szafraniec. In this paper, we consider $S$-spaces and study the structure of completely $U$-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely $U$-positive map. On the other hand, we introduce Choi $U$-matrix of a linear map and establish the equivalence of the Kraus $U$-decompositions and Choi $U$-matrices. Then we study properties of nilpotent completely $U$-positive maps. We develop the $U$-PPT criterion for separability of quantum $U$-states and discuss the entanglement breaking condition of quantum $U$-channels and explore $U$-PPT squared conjecture. Finally, we give concrete examples of completely $U$-positive maps and examples of $3 \otimes 3$ quantum $U$-states which are $U$-entangled and $U$-separable.

Mark Girard, Seung-Hyeok Kye, Erling Størmer