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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?

J Li
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  • Does this answer your question? Why is the application of probability in Quantum Mechanics fundamentally different from application of probability in other areas? – Kurt G. Mar 05 '22 at 19:19
  • @KurtG. Partially. However, the first comment under that question suggests that the answer is no? That is, “normal probability” can fully express quantum theory, even though some may prefer a different formulation? – J Li Mar 05 '22 at 19:21
  • Have you heard of Bell's inequalities? It's a theorem that local hidden variables (ie, quantities that capture the information we supposedly don't have, and which interact locally) cannot produce probability distributions like the ones predicted by quantum mechanics in carefully prepared situations. These situations have been created experimentally, and the results are completely consistent with quantum mechanics, and rule out a local hidden variable explanation. There are loopholes, but the loopholes have their own weirdnesses. – Andrew Mar 05 '22 at 20:39
  • @Andrew Yes, even though I don’t understand it super well. My follow up question is: if one is willing to give up “localness”, will this still imply that probability theory fails to describe quantum effects? – J Li Mar 05 '22 at 21:14
  • If by "epistemic" you mean that all (or a reasonably sufficient number of) possible observables of a given quantum system have a definite value assigned to them even if unknown to us (or an observer) -- therefore, the values are only revealed upon measurement -- then one can just show the impossibility of the Kolmogorov probability theory (for the quantum system representation) by reductio ad absurdum from the Kochen-Specker theorem (normally, cast in the language of projectors, but can be interpreted through the classical probability to arrive at a contradiction). – Mahir Lokvancic Mar 05 '22 at 22:34
  • E.g. https://plus.maths.org/content/john-conway-discovering-free-will-part-ii – Mahir Lokvancic Mar 05 '22 at 22:48
  • Basically, the notion of the observables as functions (or random variables) on the (Kolmogorov) probability space (or, configuration space, or phase space) inevitably leads to a contradiction, in general. (There are special cases, of course, such as a frame/observer associated with a given spectral measure can still find it useful to interpret thru the standard measure theory.) – Mahir Lokvancic Mar 05 '22 at 22:52
  • @KurtG. I think I understood better now. The idea that you can “erase” a measurement and restore interference in the double slit experiment blows my mind, and indeed seems impossible to express under classical views of probability. Thanks much for the reference. – J Li Mar 05 '22 at 23:22
  • @JLi (a) You can have non-local hidden variable theories consistent with Bell's inequalities that are deterministic but non-local. The non-localness is very hard for a physicist to accept, because special relativity implies "non-local" also means "not causal", meaning you can arrange for effects to precede causes(!) (b) Probability theory is not a dynamical theory describing how thing evolve in time, but a way of characterizing probability distributions. So you should try to be more precise; the probability distributions QM produces can be analyzed with probability theory. – Andrew Mar 06 '22 at 02:49

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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.

Peter Mortensen
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MatterGauge
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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).

anna v
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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.

WillO
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