On the structure of positive maps. II. Low dimensional matrix algebras

Abstract

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on \documentclass[12pt]{minimal}\begin{document}$M_{2} (\mathbbm {C})$\end{document}M2(C) and \documentclass[12pt]{minimal}\begin{document}$M_{3}(\mathbbm {C})$\end{document}M3(C). Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiołkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting, we show that in \documentclass[12pt]{minimal}\begin{document}$M_{2}(\mathbbm {C})$\end{document}M2(C) case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, 2Px with x maximally entangled vector and \documentclass[12pt]{minimal}\begin{document}$p\otimes \mathbb {1}$\end{document}p⊗1 with p rank 1 projector. We indicate problems with passing this result to \documentclass[12pt]{minimal}\begin{document}$M_{3}(\mathbbm {C})$\end{document}M3(C) case. Among similarities, in both \documentclass[12pt]{minimal}\begin{document}$M_{2}(\mathbbm {C})$\end{document}M2(C) and \documentclass[12pt]{minimal}\begin{document}$M_{3}(\mathbbm {C})$\end{document}M3(C) cases any unital positive map represented by self-adjoint unitary is unitarily equivalent to the transposition map. Consequently, we obtain a large family of exposed maps. Furthermore, for \documentclass[12pt]{minimal}\begin{document}$M_{3}(\mathbbm {C})$\end{document}M3(C) there appear new non-trivial class of maps represented by Choi matrices with square equal to a projector. We examine this case. We also investigate a convex structure of the Choi map, the first example of non-decomposable map. As a result the nature of the Choi map will be explained.