I noticed the question whether Bell's inequalities are based on a false premise, (https://physics.stackexchange.com/questions/219904/could-bells-theorem-be-based-on-a-false-premise) but I bring another argumentation of the question. Also I intend here to close an issue that I left open in the past, namely, whether quantum entanglements are a testimony of nonlocality. These two issues have a common ground, see below.

Bell based his inequalities on a probabilistic calculus, with positive probabilities. Is it plausible to believe that the standard formalism of QM, based on complex amplitudes, can be replaced by a formalism with real and positive numbers?

An example: the sum of two complex numbers of equal absolute value, but phases differing by π, mutually cancel out. But the sum of two probabilities never gives zero.

Proposed exercise: it is known that given a spin 1 particle one cannot construct a 101 function (see A. Peres' example with 33 directions in space - Journal of Physics A, vol. 24, Number 4, L175). Try to explain this impossibility in base of Bell's hidden variables governed by probabilities. Note that for single particles there is no question of locality or non-locality.

Then what Bell proved? Did he prove that QM is nonlocal? Or did he prove that the primary items in the quantum world are not probabilities?

This question is immediately related to another one formulated in the title, about nonlocality.

  • $\begingroup$ My understanding is that the complex plane is isomorphic to R^2, so in principle you could write such formalism in a more or less straightforward manner. $\endgroup$
    – user83548
    Nov 24, 2015 at 17:05
  • $\begingroup$ @brucesmitherson, is the isomorphism correct? My doubt comes from the fact that i times i = -1. Is this feature implemented in R^2? I am not sure. Best regards $\endgroup$
    – Sofia
    Nov 24, 2015 at 17:36
  • $\begingroup$ The easy way is to do it in matrix format en.wikipedia.org/wiki/Dual_number, but you can of course also eliminate the reference to matrices by using only products of its elements, but it will become even messier $\endgroup$
    – user83548
    Nov 24, 2015 at 17:46
  • $\begingroup$ He proved that QM is non-local. In his 1966 paper "On the problem on hidden variables in quantum mechanics", he also showed that it is possible to replace complex amplitudes by positive probabilities, showing an explicit example for the two-level system. He then also commented that the classical probability is non-local for distributed systems. $\endgroup$
    – cnguyen
    Nov 24, 2015 at 17:48
  • 1
    $\begingroup$ I advise you to take my email form the moderators While mods are able to see it, they are not permitted to give out such information. $\endgroup$
    – Kyle Kanos
    Nov 26, 2015 at 11:49

2 Answers 2


I am not a QM person, so, please bear if terminology is not correct.

In common language, what Bell's inequality achieves is -

  1. Enumerate different permutations/combinations of outcome of two particles in terms of local hidden variables, or local information plan.

  2. Consider all the permutations/combinations equally likely in a random selection experiment.

  3. Based upon permutations/combinations, establish a lower limit on percent of different outcomes. And so indirectly an upper limit on percent of same outcomes (correlation).

  4. The limits established in 3. are violated by actual experimental results.

  5. And therefore, deduces that local hidden variables are not sufficient to explain entanglement behavior.

It does not say anything about non-local, all it says is that local is not possible given 1, 2, and 3. Obviously, #2 can be the only culprit here.

It also does not say anything about multi-particle joint amplitudes.

It just disproves a local plan by establishing an arithmetic/algebraic limit which happens to mismatch the actual experimental results.

So far, it appears multi-particle joint amplitudes is responsible for the non-locality, (or the correlation in other words).

The joint amplitude functionality has to be very very complex in order to successfully work (If that is how correlation really does work). I personally tend to think that workings of nature should be simpler than that.

Joint amplitudes do not mean they are connected in any way, what it means is that they individually have enough functionality to exhibit the entanglement correlation which includes -

a) anti correlation in every direction, always

b) statistically 50/50 in any one direction for each particle independently

c) statistically Sq(sin(A/2)) correlation at relative angle of A.

d) Is there any other correlation, I do not know.

The other possibilities have not been exhausted because carrying out these experiments is not so easy. Equipment itself is not easily available, and for every variation, the equipment setup needs to be customized.


Details about this answer can be found in arXiv:physics.gen-ph/1009.2986 .

The quantum formalism works with complex amplitudes. Probabilities of detection are absolute squares of these amplitudes - i.e. secondary concepts. Typically, the formalism starts from amplitudes of local wave-packets of particles, and amplitudes of probability for joint results are obtained as products of local amplitudes. So far, the QM formalism is local.

The problem begins when to one and the same joint amplitude there are several contributions, e.g.

$$A(x=1, z=1) = A(x=1, y=1, z=1) + A(x=1, y=-1, z=1). $$


$$A(x=1, y=1, z=1) = -A(x=1, y=-1, z=1),$$ the combination of results (x=1)&(z=1) is impossible. This combination won't get out from beam-splitters, or polarizers, etc. Just imagine a man who tries to push his hands through the sleeves of a too narrow shirt. If he succeeds to introduce his right hand, the left hand won't enter the left-hand sleeve. But if he tries with the left hand and introduces it, the right hand wont's enter the right-side sleeve. So, the combination "both hands in sleeves" is impossible. The man can only remain with one hand in sleeve and one not in sleeve.

The classical world does not work with multi-particle joint amplitudes, typically it destroys their phases. These phases of multi-particle amplitudes disregard distances between particles. At the quantum level, the nature adds different contributions to a multi-particle amplitude, as if there weren't a couple of particles separated in space, but a single particle, at a single place.

  • $\begingroup$ @igael I don't know what you want to say with fewer measurements and leaving conclusion to experimenter. The problem I posed is in what concerns non-locality of QM, the Bell-type inequalities are non-conclusive. For invalidating a hypothesis, a proof has to rely only on that hypothesis, and show that a contradiction appears. Bell and his followers didn't notice that they based their inequalities on one more hypothesis, namely, that the nature works with probabilities. No, the nature works with amplitudes. So, please check if you speak of the same issue as I. Best wishes! $\endgroup$
    – Sofia
    Dec 17, 2015 at 0:58

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