Prove that maximally entangled states $|\Phi\rangle$ satisfy the identity $(U\otimes I)|\Phi\rangle=(I\otimes U^T)|\Phi\rangle$

Question

The definition of maximally entangled state is \begin{equation} \vert \Phi \rangle = \frac{1}{\sqrt{d}} \sum_i \vert i \rangle \vert i \rangle, \tag{1} \end{equation} where $d$ is the dimension of the hilbert space. Then we have the following identity \begin{equation} (U\otimes I)\vert \Phi \rangle= (I\otimes U^T)\vert \Phi \rangle, \tag{2} \end{equation} where $U$ denotes the unitary matrix.

Equation (2) is from the lower right corner at Page. 7 in Cross-Platform Verification in Quantum Networks.

My question is how to proof equation (2).

Answer 1

Start by writing $$ U=\sum_{ij}U_{ji}|j\rangle\langle i|. $$ Now evaluate $$ U\otimes I|\Phi\rangle=\frac{1}{\sqrt{d}}\sum_{ij}U_{ji}|j\rangle|i\rangle. $$ Next, realise that (again,if you don't see it,just write it out long-hand) $$ \sum_iU_{ji}|i\rangle=U^T|j\rangle, $$ so this just becomes $$ U\otimes I|\Phi\rangle=I\otimes U^T|\Phi\rangle. $$