11. As shown in Figure 3, the vertex of the parabola $y=x^{2}$ is the origin $O$, and two perpendicular lines are drawn through $O$ intersecting the parabola $y=x^{2}$ at points $P$ and $Q$, respectively. Connect $P Q$.
(1) When $P Q \parallel x$-axis, find the distance from point $O$ to $P Q$; (2) Find the maximum distance from point $O$ to $P Q$.
Three, 11. (1) When $P Q / / x$-axis, point $P$ and $Q$ are symmetric about the $y$-axis. At this time, $\triangle O P Q$ is an isosceles right triangle.
Assume point $Q(m, m)$.
Substitute into $y=x^{2}$, we get $m=1$.
Thus, $O M=m=1$.
Therefore, the distance from point $O$ to $P Q$ is 1.
(2) Assume point $P\left(a, a^{2}\right), Q\left(b, b^{2}\right)$. Then
\[
\begin{array}{l}
l_{P Q}: y=(a+b) x-a b, \\
l_{O P}: y=a x, \\
l_{O Q}: y=b x .
\end{array}
\]

Since $O P \perp O Q$, we have $a b=-1$.
Thus, $l_{P Q}: y=(a+b) x+1$.
Therefore, line $P Q$ must pass through the fixed point $M(0,1)$.
Hence, when $P Q / / x$-axis, the maximum distance from point $O$ to $P Q$ is 1.