You certainly can, but it won't be a valid signature to spend my output, only your output.

That's because it will fail the VER step (page 10 of above mentioned whitepaper) when applied to my output. It will pass the step when applied to your output.


No. The key image is an input in to the VER step, and the signature won't verify with an invalid key image. Which is to say, it can be verified. The key image also needs to be unique (not used before -- LNK step) but this only matters once the signature containing the key image is verified. It won't be if you try to use your private key and my output.

EDIT:

Oh, I think I understand what you're saying. You are saying that the verifier can't tell whether your output was spent or mine. This is true. The difference comes from the LNK step, where I'm still able to spend my output once you've done this but you can't spend yours again. That's because I can create another valid key image with any ring signature containing my output, and you can't create another valid key image for any ring signature containing yours.

So you see, my output has not been "spent" here, because I'm able to spend it later (by producing a different, but valid,  key image). Yours has been spent, because you won't be able to do this.