Sanpera–Bruß–Lewenstein Schmidt-number conjecture for PPT maps

Let $\mathcal{M}_d$ denote the algebra of complex $d\times d$ matrices, and let $T:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a linear map. Its Choi matrix is denoted by $C_T$, and $\operatorname{SN}(C_T)$ denotes the Schmidt number of that matrix. A map is completely positive and completely copositive when it and its composition with transposition are completely positive. Sanpera–Bruß–Lewenstein conjecture. If $T$ is completely positive and completely copositive, then

$$\operatorname{SN}(C_T)\leq d-1.$$

This conjecture supplies an iteration technique for studying PPT maps and entanglement breaking. The paper notes that the corresponding claim with different input and output dimensions is false, due to entangled states with positive partial transpose.