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The Born rule has been very successful in quantum mechanics. However, the interesting fact about this rule is that it only allows pairwise interference. In other words, there are no interference terms that involve three or more slits. Experiments have been conducted if this is strictly true in triple slit experiments(or five slits) and according to the results, Born's rule held within experimental error bounds.

Three Slits:

Ruling Out Multi-Order Interference in Quantum Mechanics. Urbasi Sinha et al. Science 329, 418 (2010), arXiv:1007.4193.

Five Slits:

Obtaining tight bounds on higher-order interferences with a 5-path interferometer. Thomas Kauten et al. New J. Phys. 19, 033017 (2017).

Another related article about Born rule violation is

Which-way double slit experiments and Born rule violation. J. Quach. Phys. Rev. A 95, 042129 (2017), arXiv:1610.06401.

In that paper, Dr. Quach proposed another parameter in testing the Born rule. (Past experiments used the Sorkin parameter to verify the Born rule.)

In the double slit experiment, according to Born's rule, $$\lvert A+B\rvert^2 = \lvert A\rvert^2 +\lvert B\rvert^2 + A^*B + B^*A$$

If we add another slit, according to Born's rule, $$\lvert A+B+C\rvert^2 = \lvert A\rvert^2 +\lvert B\rvert^2 + \lvert C\rvert^2+ I_{AB} + I_{AC} + I_{BC}$$

Notice that there is no interference term that involves A, B and C all at once, i.e. there is no $I_{ABC}$.

A generalized probabilistic theory might allow higher-order interference terms involving 3 or more slits and not just pairwise. What if these higher-order interference terms really exist but are somehow 'restricted' to be small, either by the set-up of the experiments, or it might be restricted by a quantum mechanical law that we don't know yet.

Question: How can we estimate the contribution of $I_{ABC}$ or higher-order interference in general?

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closed as unclear what you're asking by Emilio Pisanty, Brian Moths, Jon Custer, Cosmas Zachos, Kyle Kanos Sep 20 '17 at 9:56

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  • $\begingroup$ You don't mean $|A+B+C|^2$, do you? Otherwise, it's obvious you could not get triple products. $\endgroup$ – user154997 Sep 11 '17 at 20:29
  • $\begingroup$ @LucJ.Bourhis According to Born's Rule, triple products are not allowed but only pairwise. But what if the Born's rule is just a very good approximation and there are indeed higher-order interference terms (just like the one I proposed above) but these terms are just very small. $\endgroup$ – Anon Sep 11 '17 at 20:33
  • $\begingroup$ I_{ABC} is real, isn't it? Then a linear combination of the real part of $ABC$, $A^*BC$, $AB^*C$ and $ABC^*$? $\endgroup$ – user154997 Sep 11 '17 at 20:38
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    $\begingroup$ @J.B.Dumale Indeed the Born rule, in its initial formulation, was a wild guess, which is a justified strategy when there is experimental data to be explained and no working explanations. Are you claiming this is the case here? If so, what experimental data are left unexplained by the Born rule? If not, what kind of response are you looking for, beyond 'we agree with you in that your strategy of wild guesses is not a good idea'? $\endgroup$ – Emilio Pisanty Sep 19 '17 at 13:54
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    $\begingroup$ @J.B.Dumale I personally still don't understand why you need an $I_{ABC}$ at all. The literature you quote does provide reasons for it (the first two, to provide experimental bounds for it; the third, to understand the role of multi-slit trajectories in the path-integral formalism), but you don't posit any such foundations. I personally think in its current form this is too broad and too shapeless for this format to provide useful answers, but your edit is in review and it's not up to me. $\endgroup$ – Emilio Pisanty Sep 20 '17 at 13:25