I'd like to know how to show $$\min_{\Vert x\Vert=1=\Vert y\Vert}\left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2\geq -1/2$$$$\min_{\Vert x\Vert_2=1=\Vert y\Vert_2}\left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2\geq -1/2.$$
The inequality is discussed in a previous post Minimum of squared sum minus sum of squares but it's not clear to me how to prove this.