Fix some distribution $D$ over the unit sphere in $\mathbb{C}^n$. For $k<n$ and $x_0,x_1,\dots,x_k \overset{iid}\sim D$, call $X=[x_1,\dots,x_k]$ and identify the projection onto $\operatorname{range} X$ as $P_X = X(X'X)^{-1}X'$. I am interested in minimizing $E\left[\|P_X x_0\|^2\right]$ among distributions $D$ from some collection $\mathcal{D}$.
Examples: If $D$ is concentrated at a unit vector then $E\left[\|P_X x_0\|^2\right]=1$. On the other hand, if $D$ has randomness then $P_X$ may throw away some of $x_0$.
Finally the question: given the iid aspect, is it enough to assume $k=1$? This would allow me to work with a hopefully simpler quantity, minimizing $E\left[\|x_0'x_1\|^2\right]$.