Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following quadratic set of equations:

$ \sum_{k \neq i}^K x_k^HQ_{ik}x_k = x_i^HQ_{ii}x_i \quad i = 1, \ldots K$

$ x_i^Hx_i = 1 \quad i = 1, \ldots K$

Where $Q_{ij} \quad i=1, \ldots , K \quad j=1, \ldots , K$ are assumed to be hermitian and semidefinite positive. I do not have any clue of the direction to take, any comment is highly welcomed.

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Of course, $Q_{ij} \in \mathbb{C}^{N \times N}$ and $N > 1$