A simple question that I was pondering on while examining some algorithms that work similarly for positive definite and nonnegative matrices.

Let $\mathcal{H}$ be the space of (let's say for now $2\times 2$) Hermitian positive-definite matrices. Let $\mathcal{P}$ be the space of entrywise nonnegative matrices of the same dimension.

One can define a linear map $\Phi:\mathcal{H} \to \mathcal{P}$ in many ways; for instance, $$ \begin{bmatrix} a & b+id\\b-id & c \end{bmatrix} \mapsto \begin{bmatrix} a & a+c+b\\a+c+d & c \end{bmatrix}. $$ Is there a such a linear map that preserves matrix squaring, i.e., a map $\Phi: \mathcal{H} \to \mathcal{P}$ such that

1. $\Phi(tH)=t\Phi(H)$ for each $t \geq 0$,
2. $\Phi(H+K)=\Phi(H)+\Phi(K)$,
3. $\Phi(H^2)=\Phi(H)^2$ ?