Given a diagonal $N \times N$ matrix $\mathbf X$ whose diagonal elements are $1-i$, let
$$\mathbf Y := \mathbf X\cdot \mathbf p\cdot \mathbf p^H\cdot \mathbf X^H$$
Is it possible to set the vector $\mathbf p$ of size $N \times 1$ to have $\operatorname{tr}(\mathbf Y) = 0$, where $\mathbf p \neq \mathbf 0$?
Rephrasing and generalizing, given matrix ${\rm A} \in \Bbb C^{n \times n}$, we would like to find vector ${\rm x} \in \Bbb C^n$ such that
$$0 = \mbox{tr} \left( {\rm A} \, {\rm x} \, {\rm x}^* {\rm A}^* \right) = \mbox{tr} \left( {\rm x}^* {\rm A}^* {\rm A} \, {\rm x} \right) = {\rm x}^* {\rm A}^* {\rm A} \, {\rm x} = \| {\rm A} \, {\rm x} \|_2^2$$
where $(\cdot)^*$ denotes the Hermitian transpose. Can you take it from here?