Let's say I have a matrix,
$$ M = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} $$ I need to find a unitary matrix that takes each of the columns from this matrix, transforms it and creates a new matrix $M'$:
$$ v_1 = U \begin{bmatrix}1 \\ 0\end{bmatrix}, v_2 = U \begin{bmatrix}0 \\ 1\end{bmatrix}, M' = \begin{bmatrix}v_1 \;\;\; v_2\end{bmatrix} $$
Then, I am calculating a function involving the maximum inner product between the vectors of $M$ and vectors of $M'$. Like so:
$$ \text{max overlap} = \underset{u \in M, v \in M'}{max} -\log \langle u | v \rangle^2, $$
i.e. which pair of vectors from $M$ and $M'$ gives back the maximum overlap, then take the log squared.
I am wondering, is there a way to formulate it as an optimization problem? How do I find the optimal $U$ that maximizes this overlap? TIA.