Whether DSP_2 contains all PPT maps
Determine whether every PPT map on M_d, for d≥3, belongs to the cone DSP_2 of sums of a 2-superpositive map and the transpose of a 2-superpositive map; equivalently, establish whether every 2-positive and 2-copositive map is decomposable.
References
Furthermore, to the best of our knowledge, it is unknown whether the cone $DSP_2$ contains all PPT maps when $d\geq 3$.
Indeed, this assertion is equivalent to the dual statement \begin{equation} \label{eq:2biPTDec} \Phi \text{ is $2$-positive and $2$-copositive} \stackrel{?}{\implies} \Phi\in DEC, \end{equation} and even a stronger statement posed in , whether every $2$-entanglement-breaking map is decomposable, remains open.
We leave open the following question.
\begin{question} What is the smallest integer $k\leq d$ such that $$DSP_k\supseteq PPT?$$ \end{question}