# Implication of Born's rule on the superposition principle

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.)

• The answer by @BenCrowell answers your question perfectly so I would not add an answer but I would like to just emphasize a point. The point is to seriously accept Born's rule with respect as an axiom of quantum mechanics and not to conflate it with a separate axiom of quantum mechanics that is the principle of superposition.
– ACat
Mar 14, 2019 at 2:47
• The fact that there are no "three-modal" interference terms in the probability is a direct consequence of the Born rule being "quadratic". It doesn't imply in any sense that there are no "three-modal" superpositions. Of course, there are coherent superpositions of as many numbers of eigenstates as one wants. Out of the mere linearity of the principle of superposition, any superposition of two eigenstates automatically is a superposition of three eigenstates of a different operator, and so on.
– ACat
Mar 14, 2019 at 2:47
• How do we know that's the case? The experiment by Sinha showed (up to a certain bound) that photonic interference of any multi-modality can be decomposed into a linear combination of two-modal interferences. That means that for any single-shot, coherent experiment, a single photon could delocalize into only two modes (out of several) and yet the exact same statistics would ensue. Mar 20, 2019 at 13:00

Does the constraint of pairwise interference imply another constraint on the nature of superposition?

No. It's not a constraint on the states, it's a constraint on possible rules for how to calculate probabilities. The constraint is consistent with the Born rule (standard quantum mechanics) or with classical probability theory (probabilities always additive).

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?

Superposition, addition, and mixture here all mean the same thing, and it's not meaningful to talk about a constraint on the types of sums by saying that they have to be a certain type of sum of sums. For example, I can write $$f+g+h=(f+g/2)+(g/2+h)$$, but this is possible for any linear combination of $$f$$, $$g$$, and $$h$$.

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.)

Well, this could get bogged down in words like "really" and "choose," which we clearly can't define here, but basically no, Sorkin is talking about generalizing quantum mechanics by modifying the probability measure, not the time evolution. The time evolution according to the Schrodinger equation is such that the photon passes through all three slits. (If we were to generalize the probability measure, and wanted to retain conservation of probability, we would have to modify the dynamics somehow, but Sorkin doesn't attempt that.)

I don't think the fact that the Born rule expands into a sum of pairwise interference terms is really all that mysterious. This is simply because probabilities in quantum mechanics are proportional to the square of a wavefunction, and when you square a sum, you get a sum of second-order terms. It seems much more mysterious to me how you could get a sensible third-order version of the Born rule. E.g., it seems like phases would become observable, which creates all kinds of craziness.

• Why do you consider that superposition, addition, and mixture here all mean the same thing? This is the crux of the problem: What evidence do we have that in a coherent, single-shot experiment, we're not dealing with mixtures of pairwise superpositions? Feb 12, 2019 at 12:33

Interesting question, and QM is an interesting subject. The lack of evidence or support for a 3-modal interference term is indeed intriguing! Allow me to use a particle physics corollary: If something occurs, there must be some physical law that allows it; if something never occurs, there must be some physical law that forbids it.

I know that seems simple and perhaps too obvious, but it has been the way many things like lepton number conservation are discovered and codified.

In your question, there appears to be no evidence to support 3-modal interference, so one could conclude that there is strong support for a physical law that forbids it!

Now let's get more specific: Quantum mechanics is probabilistic, and many people use special mathematical formulas created expressly for the purpose of describing it (the delta function is a great example). We use this math because it allows us to describe something about the system, even though we do not fully understand the underlying causes. It is a common mistake to think that particles/photons/whatever are actually in two states at once. They're not. We do not understand what state a photon is in until we measure it, and to our current understanding, it is entirely probabilistic. Very clever math has been created to describe this behavior, and it is useful math because it helps us build upon our limited understanding. But do not think that a photon is actually in 2 (or 3) states at once until it is measured. We are just ignorant of which, until a measurement is made.

If a photon can take path A, B, or C, and it has some probability for each, and the interference is never 3-modal, that's weird, right? But it's a clue to something deeper and more fundamental. Why not 3-modal? Well, perhaps the fact that all interference elements show up in pairs, but each path is represented in equal amounts, is the answer. Or perhaps it has something to do with the fact that the electromagnetic force is purely polar (either positive or negative; there is no third electromagnetic mode).

I haven't researched this problem specifically, but I can tell you that my instinct lies in the fundamental laws of the EM force, or in QED. Hopefully this answer gives you a little clarity, or at least something to ponder and research more. QM and QED are still very much in need of more minds asking more interesting questions! :)

• Just one comment: There is no evidence that the interpretation of superposition as the existence of a particle in several modes at once is a mistake. If it really were a mistake, then how could you explain interaction-free measurements? Anyways, my goal is not to argue in favour of any particular interpretation, Bohmian, Copenhagen, or otherwise. Rather, it is to find the "missing link" between the fact that Born's rule precludes higher-order interference and the very nature of quantum superposition, which as as far as we know, isn't subjected to any limitations in terms of number of modes. Jul 23, 2017 at 2:53
• Well this is my point: the particle isn't in multiple modes at once, the math we use to describe its outcome only looks that way. To say it is simultaneously in multiple modes is to misinterpret the math. We use math to predict outcomes based on probability, but it is not a path descriptor like, say, a function of position over time. It is purely mathematical, not physical, until you get a result. Jul 24, 2017 at 22:25
• What evidence do we have that the particle isn't in multiple modes at once? How can one explain interaction-free measurements if there weren't for some property of the particle which pervades two modes at once? (Note: This is taking us on a tangent from the OP. My goal really wasn't to debate Bohmian vs. Copenhagen interpretations. But since we're at it, please go ahead and elaborate why you'd conclude that the particle only traverses a single mode at a time.) Jul 27, 2017 at 21:34
• "But do not think that a photon is actually in 2 (or 3) states at once until it is measured. We are just ignorant of which until a measurement is made." This is misleading. It actually is in a superposition of $2$ (or $3$) states at once until it is measured. The state collapses to one of the eigenstates of the operator that we measure but it really was in a superposition of the eigenstates before we measured it. Your view, which stands in opposition to science here, can't stand the experimental tests of even the simplest of the experiments such as the double-slit experiment.
– ACat
Mar 14, 2019 at 2:28
• Also, the link you made to the polarization states of a photon is pure numerology. The question that OP posted has nothing to do with electromagnetism (classical or quantum). The double/triple-slit experiment is a canonical experiment that works with all sorts of objects including objects with as many polarization states as one wants.
– ACat
Mar 14, 2019 at 2:31