False-positives in Bell-like Inequality Violation (CHSH specifically)

Question

Previously I asked how it makes sense for dark-counts/inefficiencies to affect loop-holes in Bell's Inequality: I have a different question:

What are all the possible false-positives in a Bell-like test? Previously I was talking about "loop-holes," but is this different from a false-positive result?

For example, if I have a generic experiment that measures the S-value for CHSH, what circumstances can lead me to a false-positive? That is, what might lead me to falsely conclude that the CHSH inequality is $>2$ and therefore nonlocal/real theories are impossible?

The only thing that I can think of is if a physical signal could travel from Alice and Bob to modify the result, if the experiment is not outside of the light-cone to prevent signaling.

Answer 1

Good follow-up question. I'd like to be sure you are on the right page before answering.

A. The CHSH inequality (and Bell inequalities in general) defines a boundary for LOCAL realistic (LR) theories, as opposed to nonlocal theories. In the most common CHSH, the boundary is maximum of 2 for LR. The theoretical QM prediction is 2√2 (which is about 2.8) as you say. The formula is designed in such a way that detector inefficiencies and the like will err on the lower side. So you won't ever see a reading exceeding 2.8, and a reading between 2 and 2.8 is fully compatible with the QM expectation. Usually, Bell tests feature readings of 2.35 and up. But keep in mind that a sufficiently high value about 2 is enough to rule out Local Realism.

Also, be aware that there are literally hundreds of different possible inequalities for quantum observables - such as time, energy, momentum - and they apply to different particle types as well. So a Bell test on photon polarization - the most common type studied - is just a fraction of the experimental canon. All of them reject Local Realism once sufficient efficiency is reached.

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B. So let's make it clear what the S>2 result means to Local Realism. It mean either Locality must be tossed out; or Realism must be tossed out; or both should be tossed out. It is not, in an of itself, a proof that the quantum world is nonlocal.

So is it possible that a: "physical signal could travel from Alice and Bob to modify the result, if the experiment is not outside of the light-cone to prevent signaling"? Well yes, except that in the Weihs Zeilinger et al experiment - and in others - the setup is designed to prevent such signaling by changing the polarization settings randomly mid-flight. But if such a signal was FTL (>c), that could explain the results. I.e. nonlocality is compatible with the results.

Once you agree that Local Realism is not tenable (as shown by literally thousands of experiments): As mentioned, you run into the issue of deciding whether to dump Locality or Realism. This is the province of the so-called Interpretations of QM. I won't go there, that is a HUGE debate subject. Keep in mind these points:

a) If you reject Realism: You end up with things like the Many Worlds Interpretation, Retrocausal/Acausal theories, and other variations that you may find unacceptable. b) If you reject Locality: You end up with Bohmian Mechanics and other nonlocal interpretations.

Regardless: No one really knows how the "magic" is pulled off in the quantum world. My next statement will be controversial to many here: "Quantum Nonlocality" (whatever that is) is now a generally accepted experimental result in the field. A search of arXiv will produce at least 5000 papers with "Nonlocal" in the title. And many others discuss rejection of Local Realism.

There are now Bell type experiments that involve remote and/or delayed choice entanglement. These are, despite the views of many, almost perfect proofs that Locality must be rejected. Again, the subject cannot be done justice in this question.

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C. And since Bell's paper appeared in 1964, important new theoretical/experimental work has been done on alternative disproofs of Local Realism. These do not use the Bell logic in any manner. Most important is the GHZ theorem. Not only does its QM prediction contradict Local Realism, it does not require a statistical boundary to do so. Each run yields either -1 or +1. The QM prediction is +1, while the LR prediction is -1. There is no gray area, it is black or white. The experimental results are a devastating disproof of LR.

EDITED TO ADD FOR THE OP:

There've been hypotheses about signals going from Alice to Bob's detectors, or from one photon to another, etc. Such being a currently unknown explanation/mechanism that might generate an S value that is artificially high. If artificially high, you are saying that could lead us to wrongly rejecting Local Realism.

When everything occurs in a common past light cone, such is perhaps a possibility. But not only are false positives needed, you actually also need false negatives to get the S value to be high too! The CHSH has 3 "+" and 1 "-" terms. How does nature know when to do err on one side (a + term) but not the other (a - term), and to err just the right statistical amount to make the QM prediction look exactly correct? The photons detected must fit within a very narrow time window, so stray photons cannot supply false coincidences.

And once again, my big question: Why would you ignore the Weihs experimental results? It was specifically designed to prevent signals from passing back and forth between Alice's side and Bob's side in such a manner as to distort the results. That was published over 25 years ago!

EDIT 2:

Also, if anyone is interested in the nuts and bolts of the Local Realistic side asking (and attempting to answer) some of the same questions you have about false/high S results:

One of the most clever groups is the team of Hans De Raedt, Kristel Michielsen, and Karl Hess, writing in various combinations. Here is a good example of their approach. Please note that their attacks on the experiments of Weihs (and others) have been well studied and refuted in the literature. So none of this should be considered accepted science.

Event-by-event simulation of Einstein-Podolsky-Rosen-Bohm experiments

There are at least a hundred other writers who are diehard Local Realists (some call them deniers). Not surprisingly, there are no experiments which have been performed by anyone which shows an actual deviation from the QM expectation values. No hidden mechanisms, hidden influences, bias, not even a hint that any of the LR hypotheses might have any support. So there's that...

Answer 2

False positives are possible even if you conduct the experiment perfectly. If a local realistic (LR) model has a nonzero chance of returning any result from any measurement, then in any finite number of trials there is a nonzero chance that it will return the same results that a quantum experiment would have, and therefore pass your test of quantumness, no matter what it may be.

That applies to all Bell-type tests. An earlier answer said of the GHZ experiment:

Each run yields either -1 or +1. The QM prediction is +1, while the LR prediction is -1. There is no gray area, it is black or white.

That's incorrect. The outcome of the (idealized) quantum GHZ experiment is +1 in every trial (in contrast to the CHSH experiment where $2\sqrt2$ is only an average), but the outcome of an experiment on a LR model may be +1 or -1. A model that just does an independent coin flip to decide the outcome of every measurement will produce +1 half the time, for an average of 0. As with the CHSH inequality, assuming LR you can prove an upper bound (½, I think?) on the average, and it's strictly less than the QM prediction, so you can statistically separate LR from QM. But it is still only statistical. You can't just do a single GHZ trial and conclude that LR is wrong, no matter how perfectly you do it.

Answer 3

I know this will be an unpopular answer/suggestion, so I have included a simple test of this, showing CHSH inequality violations still.

If we start with the approach that, in the CHSH inequality experiment, photons pairs (entangled pairs) are created with the same polarization and this is fixed at creation (not determined upon measurement), we can test what would happen. I have done this in excel, it fit the purpose and you can see what is going on.

Create a list of randomly generated photons, each with a set polarization. Simulate measurement testing, ensuring that photons are only used once and that they are randomly consumed, in no particular order, evenly divided across all detector combinations. Now the simulation can obtain a CHSH violation where S>2.

I think this violation comes about because you are combining separate, independant test results and treating them as related. In doing so, you can often exceed S value of 2.

This simulation can also give you values of S less than 2. Looking into reasoning for this, it suggests that physical experimental setup issues can cause this (false positives I guess). Simulated tests results can also reproduce these results.

In terms of a simulation, I created an excel document to do this - https://1drv.ms/x/s!Arfr_5NFNXw8aPC38X3LUGQI7oU?e=hDQTof

It is a work in progress, but is pretty much done for the most part (maybe some errors still). It currently creates 1440 photons with a set polarization angle between 0 and 360 (made setting up sheet calculations easier, I know, for instance, that 20 degree polarization is the same as 200 degrees. Measurements account for this). Random assignments of photons to detector combinations are done for testing, so 360 each (this is for even selection of detector combinations), none are reused. From here, simulated detection test results are calculated and S value generated.

If you run a simulation based on the same photons (reusing then, keeping the same polarization angles across all detector combinations), you always arrive at a value of S=2. I have included an example on a sheet in the excel document.

Seeing as the data being used in the simulation is the same to obtain S<=2 (reusing a single set of photons across detector combinations) and S>2 (merging separate test results, not reusing photon sets), does this highlight that combining separate, independent test results, can end up giving results where S>2, but not always?

Again, this is all based off an approach where we say that polarization is set at creation of the "entangled" photon pairs. They have the exact same polarization and measurement of one does not impact the other.