Considering the semidefinite relaxation technique where a quadratic program is relaxed by the full-rank semidefinite programming. This is, $x^TQx$ now reads as Tr$(XQ)$ without any rank constraint in $X$ but assuming that is semidefinite positive.
I would like to have some results regarding the quality of the aproximation.
It is clear that when the obtained matrix is rank-one, the optimal solution corresponds to the semidefinite relaxation one, but, Can we say anything about the non-rank-one solutions?
Furthermore, considering a $2 \times 2$ matrix it is not the same to have a $(1,0.2)$ , $(1,0.001)$ or $(1,0)$ eigen-decomposition. Is there any metric which provides an idea of how 'rank-one' is a matrix?
Thank you in advance.