Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, and $X_2\in\mathbb{R}^d$.
My question is: could we find a lower bound of the $(i, i)$-th element of the off-diagonal sub-matrix $X_2$, based on $\Vert X\Vert_2$, or $\Vert X_1\Vert_2$, or trace($X$), or trace($X_1$).
Any help would be highly appreciated!