# Is quantum mechanics truly probabilistic?

Probability arises inherently from a lack of information. For example, if I were to take a ball out of a bag with 3 yellow and 2 white balls, I would have a 0.6 probability of getting a yellow and a 0.4 probability of getting a white ball. However, these only apply because I cannot see where each ball is located. However, if I had the information about the forces on each ball as they were thrown into the bag, as well as the angle of my arm as I put it in, I could, without doubt, know for sure which ball I would get.

Hence, is it not possible that the entire probabilistic nature of quantum mechanics arises entirely from our lack of information on phenomenon? I have read up on the Heisenberg uncertainty principle but it seems a bit iffy that physical phenomenon are purely bounded by mathematical theory. Could there be a non-probabilistic explanation for these phenomenon?

## 3 Answers

To answer your question "Hence, is it not possible that the entire probabilistic nature of quantum mechanics arises entirely from our lack of information on phenomenon?" precisely: Yes, it is possible.

Sure, there are different ways of thinking about quantum mechanics and at some point, asking whether the randomness is true or apparent becomes a more philosophical question.

But: There is a formulation of quantum mechanics that is called Bohmian mechanics or de-Broglie-Bohm theory which is entirely deterministic. It describes, in addition to the wave function, particles which have well-defined positions. (Since the theory is non-local, it does not contradict Bell's theorem, of course.)

The probabilities that can be computed from this theory are exactly the same as in Copenhagen quantum mechanics, so it is empirically correct, but the probabilistic nature only comes from our ignorance about the initial values (initial positions of particles), as in classical physics. So to answer all claims that such a thing is impossible in principle, it just is given by this example.

• Bohmian mechanics fails to work as well as existing (non-deterministic) theories. This question and answers give more detail as to what the problems are. – StephenG Aug 17 '17 at 12:05
• Please refrain from such destructive comments which basically claim that all physics done since 1930 "fails to work", it is not the place here to discuss something like this and your position is in stark contrast to the consensus among physicists. – Luke Aug 17 '17 at 12:45
• Comments pointing out an issue with your answer are not automatically "destructive". Learn to live with them or don't post. I made no claim that "all physics done since 1930 fails to work" - it's disingenuous of you to say I did. – StephenG Aug 17 '17 at 13:02
• StephenG please read my answer to the question you've linked as there is a problem with the final paragraph of the top answer. BM works no worse (or better) than SQM. – Toby Hawkins Aug 18 '17 at 3:18
• Luke there are actually several interpretations of QM that are fully deterministic, not just BM. All but superdeterminism still require some sort of 'quantum weirdness' though (like nonlocality). – Toby Hawkins Aug 18 '17 at 3:23

It is true that many probabilities may be interpreted as descriptions of our state of knowledge. This Bayesian view of probability in laws of physics was advocated strongly by e.g. Jaynes. Bayesian probabilities satisfy Cox's axioms.

But that doesn't preclude the possibility that there exist physical probabilities that are properties of systems and the laws governing their behaviour (ontological), and not properties of our state of knowledge (epistemic).

The probabilities in quantum mechanics don't satisfy Cox's axioms. For a start, Cox's first axiom was that plausibility was described by a single real number. In quantum mechanics, a 'plausibility' is described by complex numbers, that is, pairs of real numbers.

The differences between Bayesian probability, satisfying Cox's axioms, and probabilities based on complex numbers and the amplitude-squared of their sums are exemplified by Bell's inequalities. The inequalities demonstrate that theories that represent states of knowledge with real and complex numbers result in incompatible predictions.

However, this isn't to say that one cannot make a Bayesian flavour of interpretation of complex valued states of knowledge (I already have by calling them states of knowledge). See e.g., hep-th/9307019 for a complete discussion.

Note: I had garbled my lefts and rights in the original version of this answer. I'm grateful that @Deep's comments made me realize this. It's fixed now:

Suppose we each have an urn with 3 yellow balls and 2 white balls. We take our urns into distant rooms where we can't communicate with each other. We each draw a ball. We do this a billion times. Sure enough, we each draw just about exactly 60% yellow and 40% white.

But something else odd happens: Every time we both reach into our urns with our left hands we get identically colored balls. (About 60% of the time we both get yellow and about 40% of the time we both get white.) The same is true when we use opposite hands (yours left and mine right or vice versa). But when we both reach into our urns with our right hands we frequently get opposite color balls, though we each still get a total ratio of just about 60/40. (Of course we're too far apart to be aware of this immediately, but we keep records of our draws, compare them afterward at a convenient meeting place, and this is what we find.)

Can you see how difficult that would be to explain as simple probability-as-lack-of-knowledge?

What happens in quantum mechanics is not exactly what I've just described, but something very similar, and every bit as difficult to explain on the basis of a probability-as-lack-of-knowledge model.

As @Prahar's comment says, if you want to know more, the keywords to Google for are "hidden variable" and "Bell's theorem".

• +1 Nice intuitive answer. In your example the conclusion that one may draw is that the event of drawing balls by the two observers are not independent. Statistical dependence happens all the time even in classical physics. Why should presence of statistical dependence itself rule out the probability-as-lack-of-knowledge viewpoint? – Deep Aug 17 '17 at 5:10
• @Deep: No, the non-independence is not the issue at all. Consider these four experiments: Draw a ball from Urn A with your left hand, draw a ball from Urn A with your right hand, draw a ball from Urn B with your left hand, draw a ball from Urn B with your right hand. If the outcomes of these experiments are classical random variables, they should have a joint probability distribution (whether or not they are independent). But in the above scenario,, there is NO joint probability distribution for these quantities. (Try writing one down!) – WillO Aug 17 '17 at 6:30
• Hi @WillO, thanks for the anwer :D Pardon my ignorance, but I have never heard of such a phenomenon in quantum mechanics before. Would you happen to have some links where such events occur that I could take a look at? :) – ashiswin Aug 18 '17 at 2:49
• @ashiswin: Google for "entanglement", "Bell's Theorem" and "hidden variables". Again---this is not just about lack of independence. If the only observation were that you and I always draw the same color, we could easily explain that with an "ignorance" model -- somehow the balls were arranged in advance in the urns to force us to draw this way, and we don't realize it. It's the fact that we get different results depending on whether we use the same hands or opposite hands --- that's the key to everything. – WillO Aug 18 '17 at 4:15
• @WillO You are right. This new experiment cannot be done in the classical world which does not admit non-locality. – Deep Aug 20 '17 at 5:16

## protected by Qmechanic♦Aug 17 '17 at 9:48

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