Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive semidefinite Hermitian matrix M of rank 1 such that the spectrum of $\phi(M)$ contains a negative eigenvalue?
From Choi's Theorem, I can write $$\phi(M) = \sum_i^{n^2-r}A_iMA_i^* - \sum_i^rB_jMB_j^*$$ Are the $A_i$ and $B_j$ necessarily Hermitian?
I also know that if $\phi(M)$ does not have negative eigenvalues, then it is positive semidefinite which is equivalent to being the coefficient matrix of a positive semidefinite quadratic form $f$. In this case, $f = f(x_1,...,x_n)$ can be written as the sum of squares.
Any help or advice on how to continue would be greatly appreciated.