How to use proof of lack of knowledge? This is a purely hypothetical example but is provable ignorance useful in cryptography?

For example, let's say I have a trapdoor collision resistant function. I know the trapdoor and therefore some $x_0 \neq x_1$ such that $f(x_0) = f(x_1)$. This is however, hard to find. If someone proves they know $x_0$, I can conclude that they do not know $x_1$.

Is there any context where more complicated versions of such problems is useful?
In general, you cannot prove lack of knowledge, because even if you did know something you shouldn't, you can always pretend that you don't know it and carry out the proof as if you didn't know it.



For your specific example, consider how the prover would know $x_0$.  Did you tell them what it is?  If so, that proves nothing, since they would then know $x_0$ even if they had also learned $x_1$ from somewhere else.

Conversely, if your function $f$ is collision resistant without the trapdoor, but is not injective (i.e. potential collisions do exist), then it must also be preimage resistant.  Thus, finding $x_0$ from $y = f(x_0)$ is (at least) as hard as finding $x_1$.  Thus, paradoxically, the prover exhibiting $x_0$ would in fact be evidence that they can find preimages for $f$, and thus can probably find $x_1$ as well.