Notes on extremality of the Choi map

TL;DR

This paper proves that the Choi map is extremal in the cone of all positive linear maps on 3x3 matrices, clarifying previous misconceptions and exploring related positive maps.

Contribution

It establishes the extremality of the Choi map in the full cone of positive maps on M_3, extending prior partial results and clarifying misconceptions.

Findings

Choi map is extremal in the cone of all positive linear maps on M_3.

Clarification of the correspondence between positive semi-definite biquadratic forms and positive linear maps.

Discussion of positive linear maps that agree with the Choi map on symmetric matrices.

Abstract

It is widely believed that the Choi map generates an extremal ray in the cone $\mathcal P(M_3)$ of all positive linear maps between $C^*$-algebra $M_3$ of all $n\times n$ matrices over the complex field. But the only proven fact is that the Choi map generates the extremal ray in the cone of all positive linear map preserving all real symmetric $3\times 3$ matrices. In this note, we show that the Choi map is indeed extremal in the cone $\mathcal P(M_3)$. We also clarify some misclaims about the correspondence between positive semi-definite biquadratic real forms and postive linear maps, and discuss possible positive linear maps which coincide with the Choi map on symmetric matrices.