Why can’t quantum randomness be understood as epistemic? I often hear people say that quantum randomness is “true randomness”, but I don’t really understand it. Please bear with my question.
Before the development of quantum physics, randomness is understood as being “epistemic”. That is, things appear random because we couldn’t (or haven’t yet) take a measure.
This is also how probability theory was conceptualized by Kolmogorov.
My understanding is that quantum physics can also be described using standard measure-theoretic probability theory, or, in other words, an theory with merely “epistemic” randomness.
This leads to my question/confusion: in what sense is quantum randomness non-epistemic, given it can be described by standard probability theory? Is there any property of quantum randomness that shows it cannot be epistemic?
 A: The epistemic approach to the rolling of a dice assumes a deterministic motion of the dice and the face it shows after landing. In principle we could know which face would show up. In practice this can't be known (episteme), so we assign probabilities.
In quantum mechanics, we can't know in principle. The chance has no foundation in knowledge which could be known, because there is nothing to know until the quantum dice is thrown. The outcome is a pure chance, without a mechanism steering to a determined outcome which in practice can't be known.
Einstein thought that God doesn't play dice. The outcome of a quantum mechanics measurement can't be determined by pure chance.
de Broglie introduced his pilot wave, but in Copenhagen the standard was set. Pure probability set the norm and the Born rule.
The pilot wave was furthered by Bohm in the 1950s (for which he was called many names by his contemporaries). The wave function is seen as containing the deterministic mechanism underlying the chance. Its variables though can't be local. But non-locality isn't ruled out. Because the variables are hidden, they can't be used to calculate an outcome, but in principle, a deterministic quantum mechanics dice is thrown, with non-local effects though.
A: 
My understanding is that quantum physics can also be described using standard measure-theoretic probability theory, or, in other words, an theory with merely “epistemic” randomness.

Take the simple probability of a dice to come up with one of the six numbers. In principle if the distribution from 1 to 6  is not flat, it is a true indication that there is a bias in the dice, that one could find out by measuring, weighing accurately  the dice.
The quantum mechanical probability cannot be predicted using  classical probability distributions (even convoluted ones), because it is biased by the functional dependence on the boundary conditions in the solutions of the probability distributions which  come from the quantum mechanical equations. It is the boundaries and the conservation laws that determine the probability distribution ( example).
A: There is, in general, no joint probability distribution for the outcomes of quantum measurements, which means that QM, at least as it is usually formulated, is incompatible with Kolmogorov's framework.
