It is easy to see that the real span of the set of product states is equal to the set of Hermitian operators (see e.g Can every bipartite state be written as $\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$?). My question is whether an analogous property holds for symmetric product states.
Below, $(\cdot)^*$ denotes the conjugate-transpose.
Question: Is it true that the real span
$$ S:= \text{span}_{\mathbb{R}}\{(vv^*)^{\otimes m} : v \in \mathbb{C}^d\} $$
is equal to the set of Hermitian operators whose image is contained in the symmtric space $S^m(\mathbb{C}^d)\subseteq (\mathbb{C}^d)^{\otimes m}$? In other words, is it true that for every symmetric vector $\psi \in S^m(\mathbb{C}^d)$, it holds that $\psi \psi^* \in S$?
