Ruskai–Audenaert decomposition conjecture for quantum channels

Let $P\in \operatorname{Herm}(\mathbbm{C}^{d_1}\otimes \mathbbm{C}^{d_2})$ be the Choi matrix of a completely positive trace-preserving map $T:\mathbbm{C}^{d_1\times d_1}\rightarrow \mathbbm{C}^{d_2\times d_2}$. There are $n=d_2$ positive operators $P_i$ of rank at most $r=d_1$ such that

$$P=\sum_{i=1}^n P_i$$

and either ${\rm tr}_2[P_i]\propto\1$ for all $i$, or ${\rm tr}_2[P_i]={\rm tr}_2[P_j]$ for all $i,j$. Ruskai–Audenaert conjecture. Every quantum channel admits such a decomposition; the first alternative is the weak conjecture, while the second is the strong conjecture. The weak form is equivalent to a convex decomposition into $d_2$ generalized extreme points, and the strong form additionally requires equal weights, hence a barycentric decomposition. These are open equipartition and convex-decomposition problems for completely positive trace-preserving maps.