# Is there a system where it is impossible to tell the fundamental type of probability?

Premise: What does it mean to take Planck's constant to 0? When someone takes Planck's constant to 0 then they do not effective just substitute Planck's constant with 0. The actual procedure is to create a ratio involving Planck's constant and dimensional equivalent. Now we do not vary Planck's constant to 0. Instead what is done is the physical dimensional equivalent is varied in such a way it naively appears that Planck's constant goes to 0.

There are two fundamental sources of probability I know of:

1. Semi-Classical Limit probability: I'm okay with not knowing the reason of probability - see here. The systems can it can modelled by taking Planck's constant to 0.

Note: the Born rule of quantum mechanics is not probabilistic as it was in Quantum Mechanics: Chaos and the Semi-classical Limit is the moon there when somebody is looking?

1. Quantum mechanical probability : The systems can't be modelled by effectively taking Planck's constant to 0. The notion of probability provided depends on notion of Planck's constant must be finite.

2. There are hybrid notions of probability: One can have a system where more than one fundamental source of probability is at work.1 + 2 is seen in statistical quantum mechanics with density matrices.

My question: Can one can devise a system where it is impossible to distinguish what is the type of probability is at work? (I am was hoping for something along the lines of a contrived example where finding the source of probability is uncomputable).

• 1. and 3. are both based on lack of knowledge, and 2. might be as well, if it turns out that QM does not provide a complete description, as Einstein thought. Kolmogorov's calculus is independent of the nature of probability as it is, which is one reason it is so successful. Sep 22, 2019 at 20:15
• If I ask you what is the probability of a unicorn existing outside the observable universe I'm sure we can agree that this belongs to 1. but not 3. Also I would prefer the specific argument of Einstein. A lot of the time he was talking about the measurement which is why I wrote "If this example does not work for you try Symmetry breaking." Sep 22, 2019 at 20:21
• While there are borderline cases of 1. and 3. I am open to creation of 1. + 3. = 4. But still the question of can I create a system where it is impossible to distinguish 4. and 2. ? Sep 22, 2019 at 20:24
• My point is that the distinctions themselves are based on controversial presuppositions. So, as it stands, it is "impossible to distinguish" sources of probability even in them. 1. and 3. can, perhaps, be distinguished based on our internal (computational) vs external (observational) limitations, but that is pragmatic and vague. So "borderline cases" would do just what you ask. On Einstein see e.g. Norton's site. Sep 22, 2019 at 20:46
• @Conifold I have edited the question as well to say I'm open 1 + 3 = 4 and asking if one can create an example where one cannot distinguish between 2 and 4. As for Norton's site the mentions of probability are under "God does not play dice": where he's talking about measurement consider symmetry breaking if that helps? And entanglement, EPR, etc though those arguments have been disproven by Bell. Also it's not difficult to know when the source of probability is quantum mechanical all you have to do is try to simultaneously measure non-commutating observables. Sep 22, 2019 at 20:58

As noted in the comments, tracking the real sources of randomness is controversial, and depends on metaphysical views about determinism. As we are ignorant of the "true metaphysics", one could say that our world itself is (for now) just such a system. Instead, I'll give a mathematical example. An accessible exposition of the related issues is a recent survey Computability and Randomness by Downey and Hirschfeldt.

In the theory of computability there is a notion of random sequences that can be defined in various ways. The oldest is in terms of Kolmogorov complexity (actually, we need a modification called prefix-free Kolmogorov complexity). Roughly speaking, such sequences can not be compressed, there is no algorithm generating any large segment of them that is significantly shorter than enumeration of its elements one by one. Let us assume that quantum randomness is genuine, i.e. trials of the double slit experiment generate just such incompressible sequences of 0s and 1s, and such a set-up is placed into a black box. In the other black box we have a pseudo-random (algorithmic) generator that fakes random sequences. These two black boxes will be our "system". Would we be able to tell the difference?

The obvious approach is to run statistical tests. If the sequence is truly random, 0s and 1s should occur with asymptotic frequency 1/2, and any bit string of length k should occur with asymptotic frequency 1/2 to the power k (these are called normal sequences). Moreover, any subsequence picked out by a computable function should have the same properties. These tests are known as von Mises tests, and a sequence with all computable subsequences passing them is called von Mises random. We obviously lack capacity to run all von Mises tests, let alone on all computable subsequences. But even if we could, Ville proved in 1939 that there exist von Mises random sequences that have more 0s than 1s in any initial segment. In particular, they are compressible. If our pseudo-random generator produces such sequences we wouldn't be able to tell it statistically.

To see how contrived this example is, note that the distinction only shows if all, infinitely many, terms of the sequence can be inspected, any finite bit string can be faked by a pseudo-random generator. On the other hand, Martin-Löf came up with a generalization of von Mises randomness, which is too technical to give here (very roughly, it involves testing sequences of computable subsequences), and Schnorr showed in 1973 that a sequence is incompressible if and only if it passes all Martin-Löf tests. Needless to say, the possibility of running all of those is even more remote. So even if we did have a fake random sequence presented to us in all of its infinite entirety, chances are that we wouldn't be able to expose the fakery.

• Comments are not for extended discussion; this conversation has been moved to chat. Sep 24, 2019 at 8:00

Classical physics is not the limit of quantum mechanics in which Planck's constant tends to zero. Classical behaviour, lack of quantum interference and entanglement, is a result of interactions that copy information out of a system:

https://arxiv.org/abs/quant-ph/0306072

There are quantities that can be calculated to tell whether quantum interference and entanglement are dominant effects:

https://arxiv.org/abs/quant-ph/0105072

• 1."Classical behaviour, lack of quantum interference and entanglement, is a result of interactions that copy information out of a system" 2.So when we say Planck's constant to 0 we don't mean take a dimensional unit and set it as 0. That makes no sense! What we mean is can we create a ratio such that a dimensionless quantity goes 0. In light of this I do not see 1. and 2. as incompatible. Yes there is the whole how do I get classical mechanics from quantum mechanics but taking the ratio involving plancks constant is a respectable method as well. Sep 23, 2019 at 12:27
• Also can I get a reference which explicitly states: "Classical physics is not the limit of quantum mechanics in which Planck's constant tends to zero" Sep 23, 2019 at 18:28
• I've included my references. I do not think our references are contradictory. It's the interpretation with which you use them. Sep 24, 2019 at 11:12