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$\newcommand{\diag}{\operatorname{diag}}% $The maximum distance achieved is 2, except possibly when $n=2$. By left multiplying by the unitary $U_1^{-1} \otimes U_1^{-1}$, we are reduced to $U_1 = I$ and $U_2 = U$, a unitary. Rewrite $V\otimes V - I \otimes U = (V \otimes VU^{-1} - I)(I \otimes U)$. Since the latter factor is unitary, it suffices to find $V$...

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