BACKGROUND
Born's rule quantifies the interference pattern of a single quantum particle going through two possibles paths A and B as
$P = |A|^2 + |B|^2 + ⟨A|B⟩ + ⟨B|A⟩$.
The standard interpretation of the cross-terms $⟨A|B⟩$ and $⟨B|A⟩$ is that they represent the quantum interference, i.e., the quantum correction to the classical summation of probabilities $|A|^2 + |B|^2$.
I've read a great deal about Born's rule and I'm particulalry interested in the work of Sorkin [Mod. Phys. Lett. A 9, 3119 (1994)], which asserts that no third-order (or higher) interference is allowed in quantum mechanics. I.e., consider the three-mode delocalization of a single particle into the paths A, B, and C. In this case, the interference pattern is given by
$P = |A|^2 + |B|^2 + |C|^2 + ⟨A|B⟩ + ⟨B|A⟩ + ⟨A|C⟩ + ⟨C|A⟩ + ⟨B|C⟩ + ⟨C|B⟩$.
Here again, we see the pairwise cross-terms between the different paths, but there is no higher-order terms linking A, B, and C all at once. The absence of such higher-order terms was asserted experimentally by Sinha et al. [Science 329, 418 (2010)]. Many articles have since then come to the the same conclusion that quantum interference occurs in pairs of possibilities.
QUESTION
My understanding is that quantum interference is the direct consequence of quantum (i.e., coherent) superposition. (Please correct this statement if you think it's inaccurate.) The conceptual gap that I'm trying to fill is therefore the following: Does the constraint of pairwise interference imply another constraint on the nature of superposition? I.e., are three-modal superpositions over A, B, and C, really just a mixture of pairwise suprepositions linking only---in any given coherent, single-shot experiment---modes A and B, or A and C, or B and C? To use an anthropomorphic analogy, does the photon really "split" into three paths or does it only choose two paths at a time and completely ignore the third. (Of course, we cannot tell which two it chose.)
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