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Relationship between Is the trace distance andupper bounded by the Euclidean distance?

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?

Relationship between trace distance and Euclidean distance

Suppose we have two pure state $|\psi\rangle$ and $|\phi\rangle$.

I was wondering that if the statement:

$|||\psi\rangle, |\phi\rangle||_{tr}$ is at most Euclidean distance of $|\psi\rangle$ and $|\phi\rangle$ is right or not? (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 state?

Is the trace distance upper bounded by the Euclidean distance?

Suppose we have two pure state $|\psi\rangle$ and $|\phi\rangle$.

I was wondering whether the statement: $\||\psi\rangle\!\langle\psi|- |\phi\rangle\!\langle\phi|\|_{\rm tr}$ is at most the Euclidean distance between $|\psi\rangle$ and $|\phi\rangle$. 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 states?

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Relationship between trace distance and Euclidean distance

Suppose we have two pure state $|\psi\rangle$ and $|\phi\rangle$.

I was wondering that if the statement:

$|||\psi\rangle, |\phi\rangle||_{tr}$ is at most Euclidean distance of $|\psi\rangle$ and $|\phi\rangle$ is right or not? (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 state?