# A Does the Bell theorem assume reality?

#### Demystifier

2018 Award
A large portion of physicists thinks that Bell's theorem shows that reality does not exist. Another large portion of physicists thinks that reality is not an assumption of Bell's theorem, so that Bell's theorem just proves nonlocality, period. A third large portion of physicists thinks that both reality and locality are assumptions of Bell's inequalities, so that the Bell theorem proves that either reality or locality (or both) are wrong. So who is right?

I think the best answer to this question is given by Roderich Tumulka in http://de.arxiv.org/abs/1501.04168 . Among many other papers on this subject, the Tumulka's paper stands out by clearly distinguishing 4 different notions of "reality", which he calls (R1), (R2), (R3) and (R4), and analyzing each of them separately. He concludes that only (R4) is the assumption of Bell's theorem. But he also points out that (R4) is the mildest form of realism, that it is very hard to abandon it, and that he (Tumulka) takes it for granted, just as Bell did. By taking (R4) for granted, he concludes that Bell's theorem does not assume reality.

I absolutely agree that (R4) is the only reality assumption in the Bell theorem. I also agree that it is very hard to abandon it and hence that it is quite natural to take it for granted. Nevertheless, I do not think that it is absolutely impossible to abandon it and absolutely necessary to take it for granted. Hence, for the sake of completeness, in my "Solipsistic hidden variables" paper http://de.arxiv.org/abs/1112.2034 I explore the logically consistent (even if philosophically unappealing) possibility of an interpretation in which (R4) is explicitly abandoned, to better understand in what sense nonlocality could be removed (or at least reduced) by an explicit rejection of reality.

What I would like to see a discussion about, is whether others agree that (R4) is really the only assumption of reality that is relevant to the Bell theorem and whether it is reasonable to question the validity of that assumption.

#### stevendaryl

Staff Emeritus
I absolutely agree that (R4) is the only reality assumption in the Bell theorem.
Well, the derivation of the Bell inequalities start with something like (R3).

#### DrChinese

Gold Member
The referenced paper offers some similar arguments to Norsen's line of reasoning on this subject. Essentially, that there is no realism assumption in Bell, and so you must conclude that nature is nonlocal if Bell Inequalities are violated.

On the other hand: I have said repeatedly that the realism assumption is present in Bell, it is simply not labeled as such. The realism requirement is the requirement that there is a counterfactual C in addition to A and B which can be measured. It is introduced after his (14). "Let c be a unit vector..." and works with Bell's (2).

The separability requirement is:
P(A,B|a,b,λ)=P(A|a,λ)P(B|b,λ)

Which also means:
P(A,C|a,c,λ)=P(A|a,λ)P(C|c,λ)
P(B,C|b,c,λ)=P(B|b,λ)P(C|c,λ)

But the last 2 are only good if you accept that there is (at least) an A, a B and a C - simultaneously. Which is Bell's realism. You just can't get to Bell's (15) without this assumption. It would have been better if this had been more explicit, but I guess he wrote what he thought made sense for his limited audience at the time.

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#### DarMM

I agree with @stevendaryl , Bell is assuming (R3). This is the way Leifer presents the theorem in his review paper of $\psi$-ontology theorems for example. When proved in the ontological models framework $\lambda = \psi$ is explicitly included (Beltrametti–Bugajski model, ontological framework encoding of the Dirac-VonNeumann/"Textbook" interpretation).

I think abandoning (R4) is possible, Brukner and Zeilinger's Fundamental Randomness view, QBism and Rovelli's Relational QM do it for example, although they do it in different ways from each other and your own solipsistic hidden variables model.

Linking in with our discussion on the Fraichiger-Renner paper, the typical view of the Masanes version of the result is that if you abandon (R3) you have to abandon (R4) as well, i.e. if you have no hidden variables you have to be perspectival.

#### DrChinese

Gold Member
In the referenced paper, the (R3) requirement is:
There is some (“hidden”) variable λ that influences the outcome in a probabilistic way, as represented by the probability P(A, B|a, b, λ).

But it really includes these 3 to work out in Bell - this is usually ignored but to me it is the crux of the realism assumption:

P(A,B|a,b,λ)
P(A,C|a,c,λ)
P(B,C|b,c,λ)

We are assuming the existence of a counterfactual.

#### atyy

I think the best answer to this question is given by Roderich Tumulka in http://de.arxiv.org/abs/1501.04168 . Among many other papers on this subject, the Tumulka's paper stands out by clearly distinguishing 4 different notions of "reality", which he calls (R1), (R2), (R3) and (R4), and analyzing each of them separately. He concludes that only (R4) is the assumption of Bell's theorem. But he also points out that (R4) is the mildest form of realism, that it is very hard to abandon it, and that he (Tumulka) takes it for granted, just as Bell did. By taking (R4) for granted, he concludes that Bell's theorem does not assume reality.
I think Tumulka assumes R3. However, in R3 says that $\lambda$ is not necessarily a "hidden" variable - it may be an "unhidden" variable like the wave function.

#### zonde

Gold Member
What I would like to see a discussion about, is whether others agree that (R4) is really the only assumption of reality that is relevant to the Bell theorem and whether it is reasonable to question the validity of that assumption.
Yes, R4 is the only assumption of reality that is relevant to the Bell theorem.
R3 is not a statement about reality. Bell singles out QM predictions of perfect correlations. Adding locality to that QM prediction he arrives at predeterminism of measurement outcomes and from that he concludes that "more complete specification of the state" is possible. Well, you have to express this specification in some communicable form. What possible communicable form this specification could have that would not be convertible into some set of variables?

Speaking about validity of R4, it can be questioned within philosophical discussion, but it can not be questioned within philosophical framework of scientific approach. Of course we do not believe this assumption blindly. We make plenty of redundant information and then cross check that information for consistency. From another side there is no alternative to this belief that comes even close to usefulness of this belief.

#### Demystifier

2018 Award
I think Tumulka assumes R3. However, in R3 says that $\lambda$ is not necessarily a "hidden" variable - it may be an "unhidden" variable like the wave function.
Yes, but I think that nobody argues like this: "R3 is wrong, ergo QM is local." It would be nonsense, because saying that R3 is wrong would be saying that probability is not determined by the wave function.

#### Demystifier

2018 Award
Well, the derivation of the Bell inequalities start with something like (R3).
Yes, but see my reply to atyy above.

2018 Award

#### stevendaryl

Staff Emeritus
Yes, but I think that nobody argues like this: "R3 is wrong, ergo QM is local." It would be nonsense, because saying that R3 is wrong would be saying that probability is not determined by the wave function.
So in other words, R3 is just TRUE, since there is a function that allows us to compute joint probabilities given an additional parameter (the wave function, for example).

The way that I've seen Bell's inequality derived follows along this outline
1. There is a probability distribution depending on the settings and the hidden variable $\lambda$: $P(A,B|a,b,\lambda)$.
2. (Factorizability): Assume that $P(A,B|a,b,\lambda) = P(A|a,b,\lambda) P(B|a,b,\lambda)$.
3. (Locality): Assume that $P(A|a,b,\lambda) = P(A|a,\lambda)$ and $P(B|a,b,\lambda) = P(B|b,\lambda)$
4. (Perfect correlations/anticorrelations): Perfect correlation/anti-correlation in the case where $a=b$ implies determinism.
Step 1 is just true (the example of $\lambda$ being the wave function shows that it's true). Step 3 is basically a definition of locality. Step 4 is provable. So the only controversial step is 2. Is that the assumption of realism?

#### stevendaryl

Staff Emeritus
This might be obvious, but you can show that violations of Bell's inequality don't by themselves imply nonlocality.

Here's a toy Many-Worlds type model that is local and that has the same predictions as QM for the EPR experiment. Many-Worlds itself is not actually local, in my opinion, because it relies on a wave function that is a function on configuration space, while a local realistic model would have all functions defined on physical space.

Here's how the toy local explanation for EPR works:
1. Let's define a "history" to be a sequence of records of the form $(a,A,b,B)$ where $a$ and $b$ are Alice's and Bob's chosen detector orientations, and $A$ and $B$ are their measurement results (spin-up or spin-down).
2. Let's assume a "black box" device with the following property: You input a pair of orientations, $a$ and $b$, and the device outputs either "SAME" or "DIFFERENT". The probability of the box producing "SAME" is $sin^2(\frac{\theta}{2})$ where $\theta$ is the angle between $a$ and $b$. The probability of the box producing "DIFFERENT" is $cos^2(\frac{\theta}{2})$. Boxes are identical, in that if there are two such boxes given the same inputs, they will always produce the same output. (Note this sounds like it's nonlocal, but it can be implemented using a pseudo-random number generator, if all boxes use the same generator.)
3. Let's assume that every time Alice measures her particle's spin, some strange mechanism (God maybe) creates an exact duplicate of Alice, with all of her memories. One of the copies gets result "spin-up" and the other copy gets results "spin-down". These copies of Alice do not interact with each other in any way.
4. Similarly for Bob. When he makes a measurement, two Bobs are created that get opposite results.
5. Alices and Bobs with different histories cannot interact, either.
6. The state of Alice after she has made a measurement, but before she has found out what Bob's measurement is is described by a record $(h, a, A)$, where $h$ is a history, and $a$ is her last detector orientation, and $A$ is her last result. Similarly, Bob's state before they interact is described by a record $(h,b, B)$.
7. If the black box at the current time gives "SAME" for the pair $a,b$, then Alice in state $(h, a, A)$ can only interact with Bob in state $(h, b, B)$ if $A=B$. If the black box gives "DIFFERENT", then Alice can only interact with Bob if $A \neq B$.
So in this model, what happens on Alice's end is purely local: She picks an orientation, she splits in two, and each copy gets the opposite result. Similarly, what happens on Bob's end is purely local. But if one of the Alices tries to communicate with one of the Bobs, she is only able to communicate with one that has the correct statistics. So for each of the Alices and each of the Bobs, it will seem that there is only one Alice and one Bob, and their correlations are the same as predicted by QM.

#### Demystifier

2018 Award
So the only controversial step is 2. Is that the assumption of realism?
No, it's also a part of the assumption of locality. The assumption of reality (R4) is the zeroth assumption:
0. There are measurement outcomes A and B which exist separately.

But note that if one denies 0. that A and B exist separately, then one cannot talk about separate quantities $P(A|...)$ and $P(B|...)$, in which case the controversial step 2. is wrong. So even though your 2. is not an assumption of reality per se, it is a consequence of the assumptions of reality and locality.

Finally note that some interpretations do deny 0. For instance, the relational interpretation says that A and B can only exist together, not separately, which is why the quantites $P(A|...)$ and $P(B|...)$ do not make sense. I don't think that the relational interpretation makes sense (after all, someone can measure A without measuring B), but loosely speaking that's what the relational interpretation says.

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#### Demystifier

2018 Award
So in this model, what happens on Alice's end is purely local: She picks an orientation, she splits in two
But after the splitting, the two copies don't live in the same universe. So there must exist some sort of a multiverse. The mere existence of a multiverse implies that we cannot have locality. Instead, we have to deal with some sort of multi-locality. Since there are two copies of Alice, Alice is not local but bi-local.

#### DrChinese

Gold Member
If the referenced paper - or any paper - seeks to demonstrate what the true assumptions of Bell are: it should be able to use that assumption to get to the Bell result. I don't believe that is possible with (R4). You need a counterfactual assumption of some kind.

#### Demystifier

2018 Award
The referenced paper offers some similar arguments to Norsen's line of reasoning on this subject. Essentially, that there is no realism assumption in Bell, and so you must conclude that nature is nonlocal if Bell Inequalities are violated.
That's not exactly what Tumulka says. He says that there is an assumption of realism, but that this assumption is only (R4), which is so mild that it can be taken for granted.

But the last 2 are only good if you accept that there is (at least) an A, a B and a C - simultaneously. Which is Bell's realism. You just can't get to Bell's (15) without this assumption. It would have been better if this had been more explicit, but I guess he wrote what he thought made sense for his limited audience at the time.
But this assumption of realism is the same as Tumulka's (R4), and I think that he is quite explicit that this kind of realism is assumed. To take an assumption for granted does not mean that this assumption is not used. It means that this assumption is not questioned.

#### Demystifier

2018 Award
If the referenced paper - or any paper - seeks to demonstrate what the true assumptions of Bell are: it should be able to use that assumption to get to the Bell result. I don't believe that is possible with (R4). You need a counterfactual assumption of some kind.
I believe that my analysis #13 of the @stevendaryl 's analysis #11 explains how (R4) gives the Bell result.

#### DrChinese

Gold Member
That's not exactly what Tumulka says. He says that there is an assumption of realism, but that this assumption is only (R4), which is so mild that it can be taken for granted. ...

But this assumption of realism is the same as Tumulka's (R4), and I think that he is quite explicit that this kind of realism is assumed. To take an assumption for granted does not mean that this assumption is not assumed. It means that this assumption is not questioned.
"(R4): Every experiment has an unambiguous outcome, and records and memories of that outcome agree with what the outcome was at the space-time location of the experiment."

I don't think this is realism at all. There is nothing counterfactual assumed, and is in fact the opposite of what is needed. You need an A, B and C - not all of which can be measured simultaneously - to get the Bell result.

#### zonde

Gold Member
You need an A, B and C - not all of which can be measured simultaneously - to get the Bell result.
Local model can produce predictions A, B and C without assuming anything counterfactual. You just perform three calculations with the same $\lambda$ and different measurement angles.

#### Demystifier

2018 Award
"(R4): Every experiment has an unambiguous outcome, and records and memories of that outcome agree with what the outcome was at the space-time location of the experiment."

I don't think this is realism at all. There is nothing counterfactual assumed, and is in fact the opposite of what is needed. You need an A, B and C - not all of which can be measured simultaneously - to get the Bell result.
So which of the Tumulka's (R1)-(R4) would you call realism actually needed to get the Bell result? Or do you think that we need a fifth notion of reality (R5)? If so, can you give a clear definition of (R5)?

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