It would be functionally impossible for a start. Copyright does not cover ideas and information themselves, only the form or manner in which they are expressed.
Given white noise is a random signal having equal intensity at different frequencies it's basically like saying "what's wrong with copyrighting a particular number?" it is nonsensical.
Or maybe we can all agree that just being able to describe, view, feel, smell or listen to something doesn't necessarily make it something that we can copyright or patent. We can walk back from the brink of madness towards sanity, it just takes the first few steps. :)
Clearly I didn't mean that the copyrighted work is only associated with a single number. The "two" in the "two are indistinguishable" was in response the the comment above:
> It's not the number that's copyrighted. It's the content that number represents that's copyrighted.
Which is why it is not actually in violation to have a mp3 but only to upload or play it - as that is the expression. The digital data is not actually copyrighted.
Which is, again, why white noise makes no sense to copyright - it's random noise.
Think about how many ways you can encode a single song: there are dozens of formats, each with varying bitrates and other attributes. But no matter which "number" of that song you play, a human will hear interpret them all the same.
A copyrighted file is just a large number.