Decomposability of 2-positive maps on M4

Determine whether every 2-positive linear map on M_4(\mathbb C) is decomposable, thereby extending the known M_3(\mathbb C) theorem of Yang, Leung, and Tang.

Background

The paper distinguishes its classification of rank-one determinantal kernel schemes for three-dimensional domain kernels from the operator-theoretic condition of 2-positivity, which is formulated using vectors of Schmidt rank at most two in the Choi matrix. The authors explicitly identify the decomposability of all 2-positive maps on M_4(\mathbb C) as a question beyond the results established in the paper.

The four-parameter family Ψp,q;r,t\Psi_{p,q;r,t} studied in the paper cannot resolve the question by producing a counterexample: within this family, Corollary 2-positive shows that 2-positivity is equivalent to complete positivity. Thus, whether a 2-positive but nondecomposable map exists on M_4(\mathbb C) remains outside the scope of the presented construction.

References

Two related questions lie beyond the results of this paper. The question whether every $2$-positive map on $M_4(\mathbb C)$ is decomposable extends the $M_3(\mathbb C)$ theorem of Yang, Leung, and Tang . Our $M_4$ Grassmannian result concerns #1{rank one} geometry of #1{three dimensional} domain kernels, whereas $2$-positivity is a #1{condition involving vectors of Schmidt rank at most two} on the Choi matrix; moreover Corollary~\ref{cor:2positive} shows that the family eq:Psi cannot produce a counterexample.

eq:Psi:

Ψp,q;r,t(X)=(p(x11+x22)0rx130p(x11+x22)tx32rx31tx23qx33),p,q>0,r,t≥0.\Psi_{p,q;r,t}(X)= \begin{pmatrix} p(x_{11}+x_{22})&0&r x_{13}\\ 0&p(x_{11}+x_{22})&t x_{32}\\ r x_{31}&t x_{23}&q x_{33} \end{pmatrix}, \qquad p,q>0,\quad r,t\ge0.

— Determinantal Kernel Schemes of Matrix Nets and Applications to Positive Maps  (2609.09675 - Dinh et al., 9 Sep 2026) in Section 5, “Consequences for Choi constructions and related questions”