Consider the following semidefinite program:
\begin{split} \max_{X,Y} \; & X_{12}^\top B + \mathrm{Tr}[(X_{11} + Y_{11})A]\\ \mbox{s.t.}\; & \mathrm{Tr}[X_{11} + Y_{11} - 2X_{12} E^\top - 2Y_{12} C^\top ] + d\leq 0\\ & X = \begin{bmatrix}X_{11} & X_{12}\\ X_{12}^\top & 1\end{bmatrix} \succeq 0, Y = \begin{bmatrix}Y_{11} & Y_{12}\\ Y_{12}^\top & I\end{bmatrix} \succeq 0. \end{split}
Here, $X_{12}, Y_{11}, Y_{12}\in\mathbb{R}^{n\times n}$ and $X_{12}\in\mathbb{R}^{n}$ and parameters $A, E\in\mathbb{R}^{n \times n}, B, C\in\mathbb{R}^{n}$ and $d\in\mathbb{R}$ are known. Now I want to show that the above problem is equivalent to the following quadratically constrained quadratic program (QCQP):
\begin{split} \max_{X,Y} \; & X_{12}^\top B + \mathrm{Tr}[(X_{12}X_{12}^\top + Y_{12}Y_{12}^\top)A]\\ \mbox{s.t.}\; & \mathrm{Tr}[X_{12}X_{12}^\top + Y_{12}Y_{12}^\top - 2X_{12} E^\top - 2Y_{12} C^\top ] + d\leq 0. \end{split}
To show this, I first tried using Schur complement to write the positive semidefiniteness condition as
\begin{equation} X_{11} \succeq X_{12}X_{12}^\top, \; Y_{11}\succeq Y_{12}Y_{12}^\top. \end{equation}
In the next step (which I'm not sure about), I tried using the epigraph form and replace $X_{11}$ by $X_{12}X_{12}^\top$ and similarly for $Y_{11}$. Could you explain whether the steps I used are correct and whether the two problems are really equivalent? If no, please provide conditions under which they will be equivalent.