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For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le6$, are there a $V\in\mathcal{U}\left(4\right)$ and $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$ with $\left(x_{1},x_{2}\right)\ne\left(0,0\right)$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}V\mbox{diag}\left(x_{1},x_{2},0,0\right)V^{*}\right)=0,$ $1\le i\le6$, with $x_{1}\left|u_{k,1}\right|^{2}+x_{2}\left|u_{k,2}\right|^{2}=0$ for $1\le k\le4$?

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le5$, are there $W\in\mathcal{U}\left(4\right)$ and nontrivial $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}W\mbox{diag}\left(x_{1},x_{2},0,0\right)W^{*}\right)=0,$ $1\le i\le5$, with $\left|u_{k,1}\right|^{2}x_{1}+\left|u_{k,2}\right|^{2}x_{2}=0$ for $1\le k\le4$?

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le7$, are there a $V\in\mathcal{U}\left(4\right)$ and $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$ with $\left(x_{1},x_{2}\right)\ne\left(0,0\right)$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}V\mbox{diag}\left(x_{1},x_{2},0,0\right)V^{*}\right)=0,$ $1\le i\le7$, with $x_{1}\left|u_{k,1}\right|^{2}+x_{2}\left|u_{k,2}\right|^{2}=0$ for $1\le k\le4$?

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le6$, are there a $V\in\mathcal{U}\left(4\right)$ and $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$ with $\left(x_{1},x_{2}\right)\ne\left(0,0\right)$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}V\mbox{diag}\left(x_{1},x_{2},0,0\right)V^{*}\right)=0,$ $1\le i\le6$, with $x_{1}\left|u_{k,1}\right|^{2}+x_{2}\left|u_{k,2}\right|^{2}=0$ for $1\le k\le4$?

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le7$, are there a $V\in\mathcal{U}\left(4\right)$ and $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$ with $\left(x_{1},x_{2}\right)\ne\left(0,0\right)$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}V\mbox{diag}\left(x_{1},x_{2},0,0\right)V^{*}\right)=0,$ $1\le i\le7$, with $x_{1}\left|v_{k,1}\right|^{2}+x_{2}\left|v_{k,2}\right|^{2}=0$ for $1\le k\le4$?

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le7$, are there a $V\in\mathcal{U}\left(4\right)$ and $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$ with $\left(x_{1},x_{2}\right)\ne\left(0,0\right)$, such that $\mbox{tr}\left(U_{i}\mbox{diag}\left(1,0,0,0\right)U_{i}^{*}V\mbox{diag}\left(x_{1},x_{2},0,0\right)V^{*}\right)=0,$ $1\le i\le7$, with $x_{1}\left|u_{k,1}\right|^{2}+x_{2}\left|u_{k,2}\right|^{2}=0$ for $1\le k\le4$?