"Random" is a property of your mind, not of the world. If I shuffle a deck of cards in a way that you don't know about, then the next card I pull is "random" to you even though, obviously, if one were to turn the cards face up nothing would have changed about the order of the cards, but now the next card would be fully deterministic. I could likewise sort the deck with an algorithm completely unknown to you. I could…
It has nothing to do with measurement, knowledge, randomness or even our physical reality at all. It is a much more fundamental mathematical phenomenon [0] that always occurs when you switch representation between time and frequency domains and it tells you how you can arrange information in such mixed representations.
[0]: https://en.wikipedia.org/wiki/Uncertainty_principle#Signal_p...