Let be $d\geqslant 2$. Let $\rho$ be $d\times d$ positive semidefinite matrix with $\mathrm{Tr}\left[\rho\right]=1$. Let us consider the following problem:
$$\begin{align} \max\quad&\sum_{x=1}^d\mathrm{Tr}^2\left[\rho M_x\right]\\ \text{s.t.}\quad&M_x\succeq0,\\ &\sum_{x=1}^dM_x=I\\ \end{align}$$ with $I$ being the identity. I want to find the dual of this problem. From what I gathered from other answers (notably this one and this one), the Lagrangian of this problem can be written as: $$\mathcal{L}(M, Z, \lambda) = -\sum_{x=1}^d\mathrm{Tr}^2\left[M_x\rho\right]-\sum_{x=1}^d\mathrm{Tr}\left[Z_xM_x\right]-\mathrm{Tr}\left[\left(I-\sum_{x=1}^dM_x\right)\lambda\right]$$ with $Z_x\succeq0$ and $\lambda^\dagger=\lambda$. We now solve for $\frac{\partial\mathcal{L}}{\partial M_x}=0$ (assuming I made no mistake in the differentiation): $$-2\mathrm{Tr}\left[M_x\rho\right]\rho-Z_x+\lambda=0\implies Z_x=\lambda-2\mathrm{Tr}\left[M_x\rho\right]\rho$$ We now replace this expression in the expression of $\mathcal{L}$: $$\begin{align}\mathcal{L}(M,\lambda)&=-\sum_{x=1}^d\mathrm{Tr}^2\left[M_x\rho\right]-\sum_{x=1}^d\mathrm{Tr}\left[M_x\left(\lambda-2\mathrm{Tr}\left[M_x\rho\right]\rho\right)\right]-\mathrm{Tr}\left[\left(I-\sum_{x=1}^dM_x\right)\lambda^\dagger\right]\\ &=-\mathrm{Tr}[\lambda]+\sum_{x=1}^d\mathrm{Tr}^2\left[M_x\rho\right] \end{align}$$ with $\lambda=\lambda^\dagger$ and $\lambda\succeq2\mathrm{Tr}\left[M_x\rho\right]\rho$.
And this is where I'm stuck. When I dealt with similar problems, $M$ did not intervene in the constraints. Here, the dual should be defined as the infimum over the possible $M$, but since $M$ appears in the constraint related to $\lambda$, I'm not sure about how to proceed here.
Since there is no condition on $M_x$ anymore, I'd like to say that the infimum is reached for $M_x=0$ and we would then have $\lambda\succeq0$, but I'm not sure I'm allowed to do this. Indeed, if it was possible, then the dual would be the same for this problem and for the same primal problem but without the square. Since these are two different primal problems, they shouldn't have the same dual.
What is the correct way of proceeding here?
Note that I know the optimal value of this problem, I'm only interested in the logic of deriving the dual for a more general application.
