What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. maximize over $V_1, V_2$:

$\max_{V_1, V_2} | Tr(V_1 \otimes V_2 U^{\dagger}) |$ in terms of a given $U$.

What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. minimize over $V_1, V_2$:

$\min_{V_1, V_2} | V_1 \otimes V_2 - U|$ in terms of a given $U$.

What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. minimize over $V_1, V_2$:

$\min_{V_1, V_2} | Tr(V_1 \otimes V_2 U^{\dagger}) |$ in terms of a given $U$.

What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. maximize over $V_1, V_2$:

$\max_{V_1, V_2} | Tr(V_1 \otimes V_2 U^{\dagger}) |$ in terms of a given $U$.