Separability of qubit-qudit quantum states with strong positive partial transposes

We show that all $2\otimes 4$ states with strong positive partial transposes (SPPT) are separable. We also construct a family of $2\otimes 5$ entangled SPPT states, so the conjecture on the separability of SPPT states are completely settled. In addition, we clarify the relation between the set of all $2\otimes d$separablestatesandthesetofall$2\otimes d$SPPTstatesforthecaseof$d=3,4$.