Sasho Andonov said:
I agree totally with you, but the article in Quanta Magazine gives another picture of randomness and uncertainty.
The article speaks about a more complex version of randomness arising from 50 quantum entangled particles, but essentially it is about the same notion of quantum randomness you have by measuring the spin on two particles. It may sound confusing because there isn't really a commonly agreed rigorous definition of concepts like "randomness" (or pure "chance", "unpredictability", "uncertainty", "indeterminism", etc.). That's why also there are several different randomness tests. For a non-professional audience one uses it in a somewhat loose way or defines a precise one in the context needed. The article speaks of "randomly output a binary number after being given a distribution that specifies the desired probability for each possible 50-bit output string", which, as I understand it, is a 'probability distribution'. As I said, a probability distribution like a Gaussian law is usually not connected to the concept of randomness, or at least not of pure randomness, since one event is more probable than another.
Mentz114 said:
Would a biased die whose probabilities are not all equal have non-random outcomes ?
So, for example, in this case one could say that it is not purely random, since one event is more probable than another. Another analogy could be white noise. If a signal in its spectral decomposition has every frequency as equally likely to show up, then it is 'white noise', per definition. If some part of the spectrum is more or less likely than others, then it is no longer 'white' and one does not consider it purely random. In a certain sense, one could also say that pure randomness is the maximal degree of uncertainty. But, I gave you an intuitive elementary definition of "randomness" (high school concept and which is used in most applications). If you want a more rigorous definition of randomness then you would need complexity theory which works with the entropy measure and chaos theory and that would ultimately lead to Goedel incompleteness theorem. It is much more complicated stuff (though fascinating). As an answer to your question however I believe it suffices to keep it simple.
Sasho Andonov said:
But I am a little bit sceptical about the statement that the the randomness and uncertainty are different in Quantum Physics then in other areas... If they are different why we are using the same names...?
If one believes, as the supporters of Bohmian mechanics do, that QM has hidden variables, then you are right. At bottom everything could be conceived as a deterministic process and randomness would be the same in classical as quantum physics (even though with strange pilot waves acting non-locally with 'spooky action at a distance'...). . However, I think most physicists do not support it and if one thinks of quantum indeterminism as an 'a-causal' process, then the two things are quite different. They are named with the same nomenclature only for historical reasons. To distinguish the two people speak loosely of 'classical indeterminism' vs. 'quantum indeterminism' (or randomness or uncertainty, etc.). And, most importantly, Heisenberg's uncertainty must not be connected with a measurement uncertainty arising due to the interaction of the "observer", how so many like to put it. This is a common fallacy but wrong interpretation.