On the structure of positive maps II: low dimensional matrix algebras

TL;DR

This paper investigates the structure of positive maps on low-dimensional matrix algebras, revealing differences between 2x2 and 3x3 cases using advanced tensor product and isomorphism techniques, and analyzing the convex structure of the Choi map.

Contribution

It introduces a new approach based on classical Grothendieck theorem to analyze positive maps, explaining the emergence of non-decomposable maps in 3x3 matrices.

Findings

In $M_2(C)$, positive maps can be generated from self-adjoint unitaries.

The construction used in $M_2(C)$ fails in $M_3(C)$.

The convex structure of the Choi map is elucidated, explaining its non-decomposability.

Abstract

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on $M_2(C)$ and $M_3(C)$. Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiolkowski 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 $M_2(C)$ case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, $2 P_x$ with $x$ maximally entangled vector and $p\otimes 1$ with $p$ rank 1 projector. We show why this construction fails in $M_3(C)$ case. There are also similarities. In both $M_2(C)$ and $M_3(C)$ 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. 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. This gives an information on the origin of appearance of non-decomposable maps on $M_3(C)$.