Geometric Construction of Optimal Teleportation Witnesses

Not all entangled states are useful for quantum teleportation. We present a geometric method to construct optimal teleportation witnesses, which provide operational necessary and sufficient criteria for identifying the teleportation usefulness of arbitrary two-qudit entangled states. Specifically, by developing a two-layer iterative cutting-plane algorithm to solve the shortest distance problem from the target state $\rho$ to the convex set $S$ of useless states, we obtain the projection point $\sigma^* \in S$ and then construct the optimal teleportation witness from the projection geometry. Moreover, the shortest distance $D(\rho)$ obtained during this construction also serves as a necessary and sufficient criterion for usefulness. We apply our method to identify the teleportation usefulness of three classes of entangled states.