In nielsen and chuang's QCQI book, there is a theorem called Unitary freedom in the ensemble for density matrices, which states that the sets $|\psi_i\rangle$ and $|\phi_i\rangle$ generate the same density matrix iff $$ |\psi_i\rangle=\sum_iu_{ij}|\phi_j\rangle. $$ And I also found something interesting with respect to the reduced density matrix, which I stated as a theorem while I only prove a part of it and leave another part of it as the problem of the pose:
Theorem $Tr_B|\Psi\rangle_{AB}\langle\Psi|=Tr_B|\Phi\rangle_{AB}\langle\Phi|$ iff $|\Psi\rangle_{AB}=I\otimes U|\Phi\rangle$.
It's easy to see if $|\Psi\rangle_{AB}=I\otimes U|\Phi\rangle$, then the first part of the theorem is right, while I don't know how to prove the reverse of the theorem is true. (P.S. The theorem is inspired by the unitary freedom in the ensemble of density matrices, so it might not be true.)