Let $M_n$ be the algebra of all $n$ by $n$ complex matrices. We will discuss a decomposition theorem for $k$-positive linear maps from $M_m$ to $M_n$, where $2 \leq k < \min\{m, n\}$.

Some related results on positive linear maps and implications in quantum information theory will be discussed. This talk is based on joint work, at various stages, with L.~Chen, Y. Yang and D. H. Leung.