Eylee Jung \textit{ et.al}\cite{ jung2008} had conjectured that $P_{ max}=\frac{ 1}{ 2}$ is a necessary and sufficient condition for the perfect two-party teleportation and consequently the Groverian measure of entanglement for the entanglement resource must be $\frac{ 1}{ \sqrt{ 2}}$. It is also known that prototype $W$ state is not useful for standard teleportation. Agrawal and Pati\cite{ pati2006} have successfully executed perfect (standard) teleportation with non-prototype $W$ state. Aligned with Pati's protocol\cite{ pati2006} we have considered here $Star$ type tripartite states and have shown that perfect teleportation is suitable with such states. Moreover, we have taken the linear superposition of non-prototype $W$ state and its spin-flipped version and shown that it belongs to $Star$ class. Also, standard teleportation is possible with these states. It is observed that genuine tripartite entanglement is not necessary requirement for a state to be used as a channel for successful standard teleportation. We have also shown that these $Star$ class states are $P_{ max}=\frac{ 1}{ 4}$ states and their Groverian entanglement is $\frac{ \sqrt{ 3}}{ 2}$, thus concluding that Jung conjecture is not a necessary condition.
Categories: quant-ph