With all the helps above, I found my error.
The unitary matrix $U_{AB}$ will lead to $\sum_{k}u_{ik}u_{jk}^{+}=\sigma_{ij}I_{n\times n}$.
But the computation of $Tr(o_{ij})$ results in $Tr(o_{ij})=Tr((\sum_{k}u_{jk}^{+}u_{ik})\rho_{B})\neq Tr((\sum_{k}u_{ik}u_{jk}^{+})\rho_{B})$.
So I can not get the result as $Tr(o_{ij})=\sigma_{ij}Tr(\rho_B)$.