Suppose we have two pure state $|\psi\rangle$ and $|\phi\rangle$.
I was wondering that ifwhether the statement:
$|||\psi\rangle, |\phi\rangle||_{tr}$ $\||\psi\rangle\!\langle\psi|- |\phi\rangle\!\langle\phi|\|_{\rm tr}$ is at most the Euclidean distance ofbetween $|\psi\rangle$ and $|\phi\rangle$ is right or not? (I. I know that under the qubit condition, the trace distance is exactly half of Euclidean distance.)
If it is right, then what should the vector representation be corresponding to these two pure statestates?