A Family of Completely PPT Maps

Abstract

Given $\alpha \in \mathbb{R}$ and $B_1,\dots ,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $\alpha$ and $B_1,\dots ,B_q$ are given that guarantee ${\Phi }_{\alpha ,\mathfrak{B}}^{\pm }$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.