# Ontologically speaking, does single-shot quantum interference occur in pairs of possibilities?

I've been perplexed by the semantics used in Science 329, 418-421(2010), where they state that according to

Born’s rule and its square exponent, interference always occurs in pairs of possibilities.

Alternatively,

Born’s rule predicts that quantum interference, as shown by a double-slit diffraction experiment, occurs from pairs of paths.

I understand the experiment as a whole as well as the fact that Born's rule is mathematically decomposed into pairwise terms regardless of the dimensionality of the problem. (This is presumably linked to the fact that density matrices are strictly two-dimensional although I haven't quite developed an intuition as to why that's the case.)

My problem is with the wording of those two sentences and its ontological implications. What does it mean for interference to "occur" in pairs of paths? Does it imply that for any single-shot incidence of a photon, it only chooses to delocalize into two paths at a time? Strictly speaking, interference is the result of superposition, so if interference is pairwise, does it mean that superposition also "occurs" pairwise? The semantics here are driving me crazy.

What's your interpretation (alternatively, criticism) of the authors' claim?

EDIT: I know there is a similar question, but my emphasis is on the ontological implications. As mentioned above, I already understand the formal mathematical reasons behind the term pairwise. My question is about what happens to the wave function as it traverses the interferometer (or slit) in a single-shot run.

• Does this answer your question? What are "interferences of higher order" in the context of Born rule and triple-slit diffraction? It's the fact that the absolute value of the wavefunction squared is a sum of products of pairs of possible paths, but not triplets or quadruplets of paths. The measurements in the paper check for deviations from the theoretical prediction. Commented Dec 28, 2023 at 17:21
• Based on the paper referenced above ..."pairs of paths" really means "include all combinations of pairs" so for three slits we have 3 combinations of 2 paths .... Commented Dec 28, 2023 at 17:54
• @A.P. Not quite. I got that part. It's more the ontological implications of "interference occurring in pairs of paths" that I'm asking about. Commented Dec 28, 2023 at 18:52
• Regarding the first comment by @A.P. That answer does not answer my question. I made several edits explaining why. Please re-open this question or point out where the confusion lies. Commented Dec 30, 2023 at 18:04
• In the paper, they neither claim to test what happens in a single shot nor to measure the wavefunction. That is because all they measure are photon countrates and intensities. The authors find these to confirm the predictions of standard quantum mechanics, in particular Born's rule. The statements you cite only rephrase what is mathematically explained in the first part of the paper. Commented Dec 31, 2023 at 9:23

I think the use of the term "pairwise" is a simplification.

Sticking with the YDSE, you have two slits of width $$d_1=d_2\equiv d$$ and separation (inside edge), ...um.. $$x$$.

In the big picture, the question is: does the particle go through slit 1,2, or both? And of course, the amplitude at the screen is:

$$A = A_1 + A_2$$

Pairwise. So what happens when $$x \rightarrow 0$$? Do all wave-like effects (read: diffraction) stop because we know the particle went through a slit of width $$2d$$?

No, of course not. You get a diffraction pattern that is, in the far field, the Fourier transform of the slit's aperture function.

So that's not pairwise, it's a continuous integral over all possible paths. It's always that, at least for apertures and screens.

• What if one discretizes the problem of the double slit into a simple Mach-Zehnder interferometer. That way we don't have to worry about aperture functions and the such. The question then boils down to asking whether, say, in an N-armed interferometer, the wavefunction only "traverses" two arms at a time? Commented Dec 28, 2023 at 18:54

"It's more the ontological implications of "interference occurring in pairs of paths" that I'm asking about."

"in an N-armed interferometer, the wavefunction only "traverses" two arms at a time?"

It is a mathematical convenience that we can just look at the pair combinations and calculate interference. The same would be true for water wave 3 slit inteference .... but we know the forces are actually coming from 3 slits.

Ontologically with photons we need to consider the EM field ... a field which covers all space and similarly to water, "forces" from all slits influence the interference. What we do know is there is no difference between many or single photons ... i.e. all photons act individually in the EM field. The Feynman path intergral is another method, with computers we can converge on a similar solution by considering many paths.

When we talk about photon interference the term "self interference" shows up more and more in scientific literature. It is not really that helpful of a term in explaining what is really going on .... it would be safe to say the EM field is very dynamic, it reacts to all the electrons, in the apparatus, especially excited ones in the source even before emision. Very photon has a predetermined path .... but that path can be easily modified along the way if for example the apparatus changes.

When looking at the paper in detail I'll start by mentioning some concepts about the MachZender (MZ) and about Feynman path integral. It is indeed possible to tune both arms of a typical MZ so that no light can pass to a detector ... this because the optical path lengths (in both arms) can be adjusted to 0.5 lambda multiples. (Similarily we can take a well collimated laser source and effectively turn it off by placing a perfectly aligned mirror in the path, laser current consumption drops to near zero.). Per Feynman/Dirac/others each photon determines its own path (or doesn't emit) and per the path integral if we compute many paths we see that photons like to travel path lengths of integer multiples of the wavelength. Thus we can make one arm N lambdas and the other arm N lambdas + 0.5 in length, all light travels in one arm. Per Feynman it appears as though light is resonant in its field (EM), travelling paths N lambda like in laser cavities and interference filters for example.

With respect to the experiment above, some highlights are yes signals A,B and C get back and none from E when the 2nd arm is blocked/(misaligned at F)! So my explanation is yes it would make total sense for photons to go into the second arm, bounce off the 50/50 beam splitters and return to the first arm ... so yes A,B are visible. E is not visible because .... my guess ... the defection at E is 2x, and it gets the opposite reflection/displacement when the photon comes back.

• I think the water analogy can't be extended to this, especially when talking about individual photons. Consider for example this paper: [ doi.org/10.1103/PhysRevLett.111.240402 ]. Commented Dec 29, 2023 at 10:52
• @Tfovid Looks like a neat paper but it's paywalled. Fundamentally even a single photon originates from a single electron, and that single electron has "communicated" with the entire EM field .. all space ... all slits. Even a photon in flight is able to perceive all slits ... the EM field is everywhere .... the EM field of the photon is not localized ..... until collapse/absorbtion. Commented Dec 29, 2023 at 14:33
• Here is the arXiv version: [ arxiv.org/abs/1304.7469 ]. The discussions of EM fields aren't what's at play here. Even something as simple as the Elitzur-Vaidman bomb tester can't be explained by references to the EM field. Commented Dec 29, 2023 at 17:14
• @Tfovid Thx for the paper ... interesting. I'm adding another paragraph in my answer .... not sure you'll agree with it but its my best shot. Commented Dec 29, 2023 at 22:31
• @Tfovid edited the paragraph ..... hope this helps. Commented Dec 30, 2023 at 21:07