6. A line $l$ passing through the origin intersects the hyperbola $x y=-2 \sqrt{2}$ at points $P$ and $Q$, where point $P$ is in the second quadrant. Now, fold the upper and lower half-planes to form a right dihedral angle, at which point $P$ falls to point $P^{\prime}$. Then the minimum length of the line segment $P^{\prime} Q$ is $(\quad)$.
(A) $2 \sqrt{2}$
(B) $3 \sqrt{2}$
(C) $4 \sqrt{2}$
(D) 4
6. D.

Let the equation of line $l$ be $y=k x(k<0)$.
From $x y=-2 \sqrt{2}, y=k x$, we get
$$
\begin{array}{l}
P\left(-\sqrt{\frac{-2 \sqrt{2}}{k}}, \sqrt{-2 \sqrt{2} k}\right), \\
Q\left(\sqrt{\frac{-2 \sqrt{2}}{k}},-\sqrt{-2 \sqrt{2} k}\right) .
\end{array}
$$

Draw $P^{\prime} M \perp x$-axis, and connect $M Q$.
Then $P^{\prime} M \perp M Q$.
Thus, $\triangle P^{\prime} M Q$ is a right triangle. Therefore,
$$
\begin{array}{l}
\left|P^{\prime} Q\right|^{2}=\left|P^{\prime} M\right|^{2}+|Q M|^{2} \\
=(\sqrt{-2 \sqrt{2} k})^{2}+\left(2 \sqrt{\frac{-2 \sqrt{2}}{k}}\right)^{2}+(\sqrt{-2 \sqrt{2} k})^{2} \\
=4 \sqrt{2}\left(-k-\frac{2}{k}\right) \\
\geqslant 4 \sqrt{2} \times 2 \sqrt{-k \times \frac{2}{-k}}=16 .
\end{array}
$$

Equality holds if and only if $-k=\frac{2}{-k}$, i.e., $k=-\sqrt{2}$ (discard the positive root). 
Therefore, the shortest length of $P^{\prime} Q$ is 4.