MiS Preprint Repository

Faithful teleportation of an unknown state $|\varphi ⟩$ in ${\mathbb{C}}^d$ is considered, where a general entangled state $\rho$ is served as the resource. The necessary conditions of mixed states to complete perfect teleportation are proved. Based on these results, the necessary and sufficient conditions of faithful teleportation channel $\rho$ in ${\mathbb{C}}^m\otimes {\mathbb{C}}^d$ and ${\mathbb{C}}^d\otimes {\mathbb{C}}^n$ are derived. It is shown that for $\rho$ in ${\mathbb{C}}^m\otimes {\mathbb{C}}^d$, $\rho$ must be a maximally entangled state, while for $\rho$ in ${\mathbb{C}}^d\otimes {\mathbb{C}}^n$, $\rho$ must be a mixed or pure maximally entangled state. Furthermore, the sender's measurements must be all projectors of maximally entangled states. On the other hand, for $m,n>d$, we present some classes of states for faithful teleportation.