I What is the physical significance of Bell's math?

[PeterDonis]

my equations (1)-(4) are independent.

I don't know what your equations (1)-(4) mean, because they bring in extra indexes that I don't understand. Stevendaryl's equations (1)-(4), as given in post #130, are straightforward. Do you agree with his equations (1)-(4) in post #130? If you don't, then there's where the disagreement starts.

If you do agree with his equations (1)-(4) in post #130, then he showed in post #139 how they lead to the CHSH inequality, i.e., $|g| \le 2$.

 

[PeterDonis]

Specifically, I use:

A(a,λ)=−B(a,λ)​

But that's not what you wrote earlier; in post #138, you wrote:

Using your example

A(a,λ)=−B(a,λ)=B(a,−λ′).​

The second equality is the problem; it's not right, as both I and stevendaryl have explained.

 

[stevendaryl]

The four equations, valid classically and under QM are (quoting my earlier post #133): Without any assumptions, what is actually derived from measurements is four separate expectations:

• (1) \langle a b \rangle \equiv \frac{1}{N_i} \sum^{N_i}_1 a_i b_i
• (2) \langle a b' \rangle \equiv \frac{1}{N_j} \sum^{N_j}_1 a_j b'_j
• (3) \langle a' b \rangle \equiv \frac{1}{N_k} \sum^{N_k}_1 a'_k b_k
• (4) \langle a' b' \rangle \equiv \frac{1}{N_l} \sum^{N_l}_1 a'_l b'_l

To be clear, please show how these 4 equations, in the CHSH format (which I assume is what you're using), cannot exceed 2.

I just did that! You tell me which step in the following you don't agree with:

1. For each j, -2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2
2. If it is true for each j, then it is true for the average: -2 \leq \langle a b + a b' + a' b - a' b'\rangle \leq +2
3. \langle a b + a b' + a' b - a' b'\rangle = \langle a b \rangle + \langle a b' \rangle + \langle a' b \rangle - \langle a' b' \rangle

Just to be super-clear, in the above, the notion of average is \langle Q \rangle \equiv \frac{1}{N} \sum_{n=1}^N Q_n. So there is an additional assumption that:

\frac{1}{N_1} \sum_{1,n} a_n b_n = \frac{1}{N} \sum_n a_n b_n
(where N_1 is the number of times that Alice chose to measure a and Bob chose to measure b, and \sum_{1,n} means the sum over just those values of n)

and similarly for the other averages. The possibility that that's not the case is one of the loopholes that I mentioned in a previous post. For the average of the quantity a_n b_n to depend on choices made by Alice and Bob would be a very strange situation. It's comparable to the following:

I hide 100 balls in 5000 boxes, all of which look identical on the outside. So on the average, 1 out of 50 boxes contains a ball. Then I ask you to choose 1000 boxes. You would expect that 1 in 50 of those would contain a ball, that the ratios are the same as for the full set. That isn't a logical necessity, but it is what people normally assume when they use sampling to get an idea about the likelihood of something.

 

[Mentz114]

@N88, I think I understand your problem with this.

$-2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2$

This is only a mathematical identity if $a, a', b, b'$ is the exactly the same thing in each term. This is not possible in practical terms so one must assume that the identical preparation and many repetitions is enough for this to hold within a small margin.

We could write
$-2 \leq a_j b_j + \bar{a}_j b_j' + a_j' \bar{b}_j - \bar{a}_j' \bar{b}_j' \leq +2$

is true iff all barred things belong to the same equivalence class as the unbarred things.

 

[rubi]

For each j, -2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2

This step doesn't work in N88's case, since you don't know that the $a_j$ in the first term is the same as the $a_j$ in the second term. These $a_j$'s come from different sequences of measurements. What you really measure is a huge list $(a_j,b_j,\alpha_j,\beta_j)$ (the experimenter has to memorize the angle settings) and the individual correlations are given by $C_{\alpha\beta}=\frac 1 {N(\alpha,\beta)} \sum _{\alpha_j=\alpha,\beta_j=\beta} a_j b_j$, i.e. you perform a sum over subsequences. For example the first $j$ such that $\alpha_j=\alpha$ and $\beta_j=\beta$ might be $j=3$ and the first $j$ such that $\alpha_j=\alpha$ and $\beta_j=\beta'$ might be $j=5$. In order to apply your inequality to the sum $C_{\alpha\beta} + C_{\alpha\beta'} + \cdots$, you need that $a_3 = a_5$, because this was used in the proof of the bound of your inequality ($a_j b_j + a_j b_j' + \cdots = a_j(b_j+b_j') + \cdots$). If you don't make such an assumption, the bound will be $4$ instead of $2$.

Your proof applies to the situation, where one measures a list $(a_j,b_j,a_j',b_j')$ of 4 spins. But in a Bell test experiment, one measures 2 spins and 2 angles instead. In order to map your proof onto the situation of a real Bell test experiment, you need assumptions on the sequence $(a_j,b_j,\alpha_j,\beta_j)$.

 

[Mentz114]

..
..
In order to map your proof onto the situation of a real Bell test experiment, you need assumptions on the sequence $(a_j,b_j,\alpha_j,\beta_j)$.

One assumption that is required is the the sequence $a$ and $\bar{a}$ etc ( see my post above) must contain the same number of 1's. Sounds unlikely but this property alone sets the correlation limits between any sequences.

 

[N88]

I just did that! You tell me which step in the following you don't agree with:

1. For each j, -2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2
2. If it is true for each j, then it is true for the average: -2 \leq \langle a b + a b' + a' b - a' b'\rangle \leq +2
3. \langle a b + a b' + a' b - a' b'\rangle = \langle a b \rangle + \langle a b' \rangle + \langle a' b \rangle - \langle a' b' \rangle

Just to be super-clear, in the above, the notion of average is \langle Q \rangle \equiv \frac{1}{N} \sum_{n=1}^N Q_n. So there is an additional assumption that:

\frac{1}{N_1} \sum_{1,n} a_n b_n = \frac{1}{N} \sum_n a_n b_n
(where N_1 is the number of times that Alice chose to measure a and Bob chose to measure b, and \sum_{1,n} means the sum over just those values of n)

and similarly for the other averages. The possibility that that's not the case is one of the loopholes that I mentioned in a previous post. For the average of the quantity a_n b_n to depend on choices made by Alice and Bob would be a very strange situation. It's comparable to the following:

I hide 100 balls in 5000 boxes, all of which look identical on the outside. So on the average, 1 out of 50 boxes contains a ball. Then I ask you to choose 1000 boxes. You would expect that 1 in 50 of those would contain a ball, that the ratios are the same as for the full set. That isn't a logical necessity, but it is what people normally assume when they use sampling to get an idea about the likelihood of something.

You write: For each j, -2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2. (SD-1)

But for the 4 equations that I gave you, the fundamental expression is:

|a_i b_i + a_j b_j' + a_k' b_k - a_l' b_l'| \leq +4. (N88-1)

Please note, term by term, that you did not do what was requested. To be clear, my four equations, valid classically and quantum mechanically, follow. Without any assumptions, they represent what is actually derived from measurements; four separate expectations:

• (1) \langle a b \rangle \equiv \frac{1}{N_i} \sum^{N_i}_1 a_i b_i
• (2) \langle a b' \rangle \equiv \frac{1}{N_j} \sum^{N_j}_1 a_j b'_j
• (3) \langle a' b \rangle \equiv \frac{1}{N_k} \sum^{N_k}_1 a'_k b_k
• (4) \langle a' b' \rangle \equiv \frac{1}{N_l} \sum^{N_l}_1 a'_l b'_l

Please compare -- term by term -- your requirement in the heart of (SD-1) to my (N88-1); which is valid valid classically and quantum mechanically. In my terms, your equations would be valid for an ordered ensemble of classical objects that could be repeatedly tested in the same order. But (it seems to me) such objects would have no place under EPRB (Bell 1964).

So my question remains: What is the physical significance of the changes that you require for (SD-1) to be valid, please?

 

[N88]

@N88, I think I understand your problem with this.

$-2 \leq a_j b_j + a_j b_j' + a_j' b_j - a_j' b_j' \leq +2$

This is only a mathematical identity if $a, a', b, b'$ is the exactly the same thing in each term. This is not possible in practical terms so one must assume that the identical preparation and many repetitions is enough for this to hold within a small margin.

We could write
$-2 \leq a_j b_j + \bar{a}_j b_j' + a_j' \bar{b}_j - \bar{a}_j' \bar{b}_j' \leq +2$

is true iff all barred things belong to the same equivalence class as the unbarred things.

Thanks for this; I believe you do understand my problem. I trust the reply just posted (for stevendaryl) shows how my equations make your point. Critical comments on such a comparison would be welcome: but I suspect the equivalence class would need to be be the smallest such: an equality relation.

 

[Mentz114]

Thanks for this; I believe you do understand my problem. I trust the reply just posted (for stevendaryl) shows how my equations make your point. Critical comments on such a comparison would be welcome: but I suspect the equivalence class would need to be be the smallest such: an equality relation.

Thanks. [ I deleted some nonsense here].

The point is that this is a classical theorem and QM does seem to break it.

 

[N88]

Thanks. But it is possible that an experiment will ( very nearly) satisfy the conditions that the limit theorem requires. It is a loophole that ( to my knowledge) is not tested. One assumption is that all subsequences appear random - but I'm not sure what random means in this context.

I'm not at all sure re the possibility that you raise; and I'm not sure that rubi's careful checking could do the job, except with classical objects. I believe the facts go this way:

(SD-1) will be satisfied by classical objects, as suggested earlier. (N88-1) will provide a limit of 2 for classical objects and limit of 2√2 for quantum-entangled objects (eg, EPRB).

 

[Mentz114]

I'm not at all sure re the possibility that you raise; and I'm not sure that rubi's careful checking could do the job, except with classical objects. I believe the facts go this way:

(SD-1) will be satisfied by classical objects, as suggested earlier. (N88-1) will provide a limit of 2 for classical objects and limit of 2√2 for quantum-entangled objects (eg, EPRB).

I deleted the offending stuff before you posted - sorry about the confusion.

 

[Jilang]

Isn't the difference just down to the fact that identically prepared classical entities will share the same $\lambda$ but identically prepared quantum mechanical entities won't?

 

[stevendaryl]

Thanks for this; I believe you do understand my problem. I trust the reply just posted (for stevendaryl) shows how my equations make your point. Critical comments on such a comparison would be welcome: but I suspect the equivalence class would need to be be the smallest such: an equality relation.

Let me try one more time, and make it super concrete. Suppose that we repeatedly do the following 4 measurements:

1. We produce a correlated pair. Alice measures the spin of her particle along axis a. Bob measures along axis b
2. We produce a correlated pair. Alice measures along a, Bob measures along b'
3. We produce a correlated pair. Alice measures a'. Bob measures b
4. We produce a correlated pair. Alice measures a'. Bob measures b'

We do these four things over and over, N times. (So we actually produce 4N correlated pairs)

Then we compute:
C(a,b) = \frac{1}{N} \sum_n a_n b_n (where n ranges over 1, 5, 9, etc.)
C(a,b') = \frac{1}{N} \sum_n a_n b_n' (where n ranges over 2, 6, 10, etc.)
C(a',b) = \frac{1}{N} \sum_n a_n' b_n (where n ranges over 3, 7, 11, etc.)
C(a', b') = \frac{1}{N} \sum_n a_n' b_n' (where n ranges over 4, 8, 12, etc.)

Now, the hidden-variable assumption is this: Although

• nobody measured a_n b_n when n=2, 3, 4, 6, 7, 8, 10, 11, 12...
• nobody measured a_n b_n' when n=1, 3, 4, 5, 7, 8, 9, 11, 12...
• nobody measured a_n' b_n when n=1, 2, 4, 5, 6, 8, 9, 10, 12...
• nobody measured a_n' b_n' when n=1, 2, 3, 5, 6, 7, 9, 10, 11...

Those variables had definite values. So even though we don't know what the values were for some variables on some rounds, it makes sense to talk about the following averages:

1. D(a,b) = \frac{1}{4N} \sum_n a_n b_n
2. D(a,b') = \frac{1}{4N} \sum_n a_n b_n'
3. D(a',b) = \frac{1}{4N} \sum_n a_n' b_n
4. D(a',b') = \frac{1}{4N} \sum_n a_n' b_n'

where this time, all sums extend over all values of n from 1 to 4N.

The assumption is that

1. D(a,b) \approx C(a,b)
2. D(a, b') \approx C(a, b')
3. D(a', b) \approx C(a', b)
4. D(a', b') \approx C(a', b')

That is, the assumption is that the unmeasured quantities have the same statistical properties as the measured quantities. In the limit as N \rightarrow \infty, it is assumed that the averages C approach the averages D.

 

[stevendaryl]

Isn't the difference just down to the fact that identically prepared classical entities will share the same $\lambda$ but identically prepared quantum mechanical entities won't?

No, there is no assumption classically that identically prepared systems will share the same \lambda, only that identically prepared sequences of systems will share the same distribution on possible values of \lambda.

 

[rubi]

it's certainly possible to fail to recognize that the stricter inequality is true.

The stricter inequality might be true, but it can't be applied in the case N88 is talking about. If you are in the situation of a Bell test experiment, where you recorded the sequence $(a_j,b_j,\alpha_j,\beta_j)$ of 2 spins and 2 angles, you just can't apply the stricter inequality to the sum $C_{\alpha\beta}+C_{\alpha\beta'}+C_{\alpha'\beta}-C_{\alpha'\beta'}$ and hence, you won't obtain the a bound of $2$. In fact, in this situation, the bound $2$ is false and can be violated easily. Here's a list that violates it: $(1,1,\alpha,\beta)$, $(1,1,\alpha,\beta')$, $(1,1,\alpha',\beta)$, $(1,-1,\alpha',\beta')$. In that case, the sum will be equal to $4$.

You obtain the bound of $2$ only in the hypothetical situation in which CFD is not violated and you can assume that your recorded sequence $(a_j,b_j,a_j',b_j')$ consists of 4 spins in each run (even though you measure only 2 of them).

The assumption is that

1. D(a,b) \approx C(a,b)
2. D(a, b') \approx C(a, b')
3. D(a', b) \approx C(a', b)
4. D(a', b') \approx C(a', b')

That is, the assumption is that the unmeasured quantities have the same statistical properties as the measured quantities. In the limit as N \rightarrow \infty, it is assumed that the averages C approach the averages D.

In the counterfactually definite situation, this can even be proved. However, if CFD is violated, then there is no reason to expect something like this to be true. The quantity $D$ can't even be meaningfully defined.

 

[stevendaryl]

Let me try one more time, and make it super concrete. Suppose that we repeatedly do the following 4 measurements:

1. We produce a correlated pair. Alice measures the spin of her particle along axis a. Bob measures along axis b
2. We produce a correlated pair. Alice measures along a, Bob measures along b'
3. We produce a correlated pair. Alice measures a'. Bob measures b
4. We produce a correlated pair. Alice measures a'. Bob measures b'

We do these four things over and over, N times. (So we actually produce 4N correlated pairs)

Then we compute:
C(a,b) = \frac{1}{N} \sum_n a_n b_n (where n ranges over 1, 5, 9, etc.)
C(a,b') = \frac{1}{N} \sum_n a_n b_n' (where n ranges over 2, 6, 10, etc.)
C(a',b) = \frac{1}{N} \sum_n a_n' b_n (where n ranges over 3, 7, 11, etc.)
C(a', b') = \frac{1}{N} \sum_n a_n' b_n' (where n ranges over 4, 8, 12, etc.)

Now, the hidden-variable assumption is this: Although

• nobody measured a_n b_n when n=2, 3, 4, 6, 7, 8, 10, 11, 12...
• nobody measured a_n b_n' when n=1, 3, 4, 5, 7, 8, 9, 11, 12...
• nobody measured a_n' b_n when n=1, 2, 4, 5, 6, 8, 9, 10, 12...
• nobody measured a_n' b_n' when n=1, 2, 3, 5, 6, 7, 9, 10, 11...

Those variables had definite values. So even though we don't know what the values were for some variables on some rounds, it makes sense to talk about the following averages:

1. D(a,b) = \frac{1}{4N} \sum_n a_n b_n
2. D(a,b') = \frac{1}{4N} \sum_n a_n b_n'
3. D(a',b) = \frac{1}{4N} \sum_n a_n' b_n
4. D(a',b') = \frac{1}{4N} \sum_n a_n' b_n'

where this time, all sums extend over all values of n from 1 to 4N.

The assumption is that

1. D(a,b) \approx C(a,b)
2. D(a, b') \approx C(a, b')
3. D(a', b) \approx C(a', b)
4. D(a', b') \approx C(a', b')

That is, the assumption is that the unmeasured quantities have the same statistical properties as the measured quantities. In the limit as N \rightarrow \infty, it is assumed that the averages C approach the averages D.

So the proof of Bell's inequality is a proof about the averages D. What we actually measure is a different kind of average, C. So that's one of the loopholes for Bell's theorem--maybe for some reason the averages C are not equal to the averages D, and so the inequalities don't apply to the measured averages C.

@N88 is correct, that unless you assume that the averaging process C gives approximately the same result as the theoretical averages D, then you can't prove Bell's inequality, and in fact, you have a much weaker inequality:

-4 \leq C(a,b) + C(a, b') + C(a', b) - C(a', b') \leq 4

 

[N88]

So the proof of Bell's inequality is a proof about the averages D. What we actually measure is a different kind of average, C. So that's one of the loopholes for Bell's theorem--maybe for some reason the averages C are not equal to the averages D, and so the inequalities don't apply to the measured averages C.

@N88 is correct, that unless you assume that the averaging process C gives approximately the same result as the theoretical averages D, then you can't prove Bell's inequality, and in fact, you have a much weaker inequality:

-4 \leq C(a,b) + C(a, b') + C(a', b) - C(a', b') \leq 4

Thanks for going "super-concrete". I trust I have it right: that C denotes the measured correlations for EPRB. So, continuing to be super-concrete under EPRB (Bell 1964), and consistent with QM theory and QM calculations:

C(a,b) = \frac{1}{N} \sum_n a_n b_n (where n ranges over 1, 5, 9, etc.) = -a\cdot b.
C(a,b') = \frac{1}{N} \sum_n a_n b_n' (where n ranges over 2, 6, 10, etc.) = -a\cdot b'.
C(a',b) = \frac{1}{N} \sum_n a_n' b_n (where n ranges over 3, 7, 11, etc.) = -a'\cdot b.
C(a', b') = \frac{1}{N} \sum_n a_n' b_n' (where n ranges over 4, 8, 12, etc.) = -a'\cdot b'.

Re D, I'll have more to say soon.

Thanks again.

 

[Jilang]

So these are different sets? The Venn diagram that is often shown to explain the issue cannot apply then?
http://theory.physics.manchester.ac.uk/~judith/AQMI/PHYS30201se24.xhtml

 

[N88]

So these are different sets? The Venn diagram that is often shown to explain the issue cannot apply then?
http://theory.physics.manchester.ac.uk/~judith/AQMI/PHYS30201se24.xhtml

For me, the answers to your questions are Yes and Yes. The following article by d'Espagnat (1979) has Bell's (1980) endorsement. You will see the above Venn diagram developed there.

http://www.scientificamerican.com/media/pdf/197911_0158.pdf

General note: On p.158 we see these principles of local realism: (i) realism -- regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers; (ii) locality -- no influence of any kind can propagate superluminally; (iii) induction -- legitimate conclusions can be drawn from consistent observations.

On p.166, in the last paragraph and continuing into p.167, we see d'Espagnat's subtle (but false) inference: that is, he ignores consistent observations re the validity of QM and the repeated verification of Bohr's well-known insight (that a measurement perturbs the measured system). Thus, in my view, the above principles of local realism remain valid.

So this is not a dispute about principles differing from the local realism of Bell and d'Espagnat. For me it's a lesson on the need to infer correctly to quantum-mechanically validated experimental results.

 

[N88]

Let me try one more time, and make it super concrete. Suppose that we repeatedly do the following 4 measurements:

1. We produce a correlated pair. Alice measures the spin of her particle along axis a. Bob measures along axis b
2. We produce a correlated pair. Alice measures along a, Bob measures along b'
3. We produce a correlated pair. Alice measures a'. Bob measures b
4. We produce a correlated pair. Alice measures a'. Bob measures b'

We do these four things over and over, N times. (So we actually produce 4N correlated pairs)

Then we compute:
C(a,b) = \frac{1}{N} \sum_n a_n b_n (where n ranges over 1, 5, 9, etc.)
C(a,b') = \frac{1}{N} \sum_n a_n b_n' (where n ranges over 2, 6, 10, etc.)
C(a',b) = \frac{1}{N} \sum_n a_n' b_n (where n ranges over 3, 7, 11, etc.)
C(a', b') = \frac{1}{N} \sum_n a_n' b_n' (where n ranges over 4, 8, 12, etc.)

Now, the hidden-variable assumption is this: Although

• nobody measured a_n b_n when n=2, 3, 4, 6, 7, 8, 10, 11, 12...
• nobody measured a_n b_n' when n=1, 3, 4, 5, 7, 8, 9, 11, 12...
• nobody measured a_n' b_n when n=1, 2, 4, 5, 6, 8, 9, 10, 12...
• nobody measured a_n' b_n' when n=1, 2, 3, 5, 6, 7, 9, 10, 11...

Those variables had definite values. So even though we don't know what the values were for some variables on some rounds, it makes sense to talk about the following averages:

1. D(a,b) = \frac{1}{4N} \sum_n a_n b_n
2. D(a,b') = \frac{1}{4N} \sum_n a_n b_n'
3. D(a',b) = \frac{1}{4N} \sum_n a_n' b_n
4. D(a',b') = \frac{1}{4N} \sum_n a_n' b_n'

where this time, all sums extend over all values of n from 1 to 4N.

The assumption is that

1. D(a,b) \approx C(a,b)
2. D(a, b') \approx C(a, b')
3. D(a', b) \approx C(a', b)
4. D(a', b') \approx C(a', b')

That is, the assumption is that the unmeasured quantities have the same statistical properties as the measured quantities. In the limit as N \rightarrow \infty, it is assumed that the averages C approach the averages D.

Continuing your appreciated "super-concrete" theme: Two posts above, I agreed with your analysis re C and gave what I believe to be results agreed by us jointly. When it comes to D, imho we are dealing with counterfactuals; ie, counterfactual statements of the "IF this … THEN this" kind. Note that counterfactuals are NOT contrary (fake) facts. They are facts about what would have happened IF we had done something different.

So, IF we a dealing with quantum objects (where a decoherent interaction occurs), THEN only one of your four D examples can be valid. Thus, in comparison with the C factors:

1. IF we had conducted D-1 under EPRB, THEN the result would have been: D(a,b) = C(a,b).
2. IF we had conducted D-2 under EPRB, THEN the result would have been: D(a,b') = C(a,b').
3. IF we had conducted D-3 under EPRB, THEN the result would have been: D(a',b) = C(a',b).
4. IF we had conducted D-4 under EPRB, THEN the result would have been: D(a',b') = C(a',b').

However, IF you wish to insist that all four D examples are jointly permissible: well then (in my view) you are (in fact) invoking ordered and unperturbed classical objects. Thus, whereas the preceding EPRB results would have delivered a maximised CHSH result of 2√2, your "jointly valid" and therefore classical D examples would deliver a maximised CHSH result of 2.

My above reply to Jilang explains, for me, the departure of Bell's analysis from local-realism in a quantum setting; ie, from the EPRB setting that is the focus of Bell (1964). For me, the above (i)-(iii) Bell/d'Espagnat principles of local-realism "should" be acceptable to all local realists.

 

[stevendaryl]

Continuing your appreciated "super-concrete" theme: Two posts above, I agreed with your analysis re C and gave what I believe to be results agreed by us jointly. When it comes to D, imho we are dealing with counterfactuals; ie, counterfactual statements of the "IF this … THEN this" kind. Note that counterfactuals are NOT contrary (fake) facts. They are facts about what would have happened IF we had done something different.

The realist assumption is that the variables a_n, a_n', b_n, b_n' exist whether we measure them or not. So under this assumption, the averages D are not counterfactual---they are actual averages of unmeasured quantities.

 

[stevendaryl]

In the counterfactually definite situation, this can even be proved. However, if CFD is violated, then there is no reason to expect something like this to be true. The quantity $D$ can't even be meaningfully defined.

We're talking in the context of Bell's inequality, which was derived under the assumption that we had a local realistic theory. With such a theory, the variables a_n, a_n', b_n, b_n' exist independently of whether anybody measures them, and so it makes sense to talk about averages of them.

You can certainly reject that assumption, which means rejecting the assumption that Bell proved false, anyway.

 

[Jilang]

The realist assumption is that the variables a_n, a_n', b_n, b_n' exist whether we measure them or not. So under this assumption, the averages D are not counterfactual---they are actual averages of unmeasured quantities.

Are we able to assume that realistic properties do exist independent of measurement, but that the act of measurement changes their distribution?

 

[DrChinese]

Are we able to assume that realistic properties do exist independent of measurement, but that the act of measurement changes their distribution?

That's not possible if there are to be perfect correlations. In effect, the measurement apparatus itself cannot be more than a static participant (i.e. precisely the same impact on both sides). Else there would be some variability introduced at Alice or Bob's setups. And the outcomes would not match, as they actually do.

 

[stevendaryl]

Are we able to assume that realistic properties do exist independent of measurement, but that the act of measurement changes their distribution?

That's certainly a logical possibility, but there is no way to explain the perfect anti-correlations achieved in EPR that way.

 

[Jilang]

We've had discussions here before which would appear to suggest the perfect anti correlations are not the problem. I am thinking of the "toy model" where anything in the upper hemisphere registers up etc. The tricky part is the correlation between the smaller angles.

 

[DrChinese]

We've had discussions here before which would appear to suggest the perfect anti correlations are not the problem. I am thinking of the "toy model" where anything in the upper hemisphere registers up etc. The tricky part is the correlation between the smaller angles.

And again, as before, there aren't any such realistic models outside of ones in which there are nonlocal effects.

 

[N88]

The realist assumption is that the variables a_n, a_n', b_n, b_n' exist whether we measure them or not. So under this assumption, the averages D are not counterfactual---they are actual averages of unmeasured quantities.

As I wrote above: it would never occur to me to make such an assumption under EPRB! So I am inclined to say that this statement should be known as the 'strawman-realist' assumption.

As I have accepted from my first encounter with Bell, and as you advised rubi: "You can certainly reject that assumption, which means rejecting the assumption that Bell proved false, anyway."

1: For me, that is one of the most helpful comments that I have ever seen on Bell'sTheorem.​

For I am a realist under this article by d'Espagnat (which Bell endorsed): http://www.scientificamerican.com/media/pdf/197911_0158.pdf

2: On p.158 we see these principles of local realism: (i) realism -- regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers; (ii) locality -- no influence of any kind can propagate superluminally; (iii) induction -- legitimate conclusions can be drawn from consistent observations.​

If we now return to the OP, can we now say with total accuracy:

3: In Bell's (1964) move from his equation (14a) to (14b), Bell's use of [A(b,λ)]² = 1 (equation (X) in the OP) is his way of setting up a "strawman-realist" assumption for mathematical consideration (and rejection)?​

Note: This then leaves me wondering why the general claim associated with Bell's theorem is that "local-realism" is refuted? I can accept that local strawman-realism is refuted. But realism under Bohr's insight (that a measurement may perturb the measured system) is not refuted.

 

[N88]

If you don't share the definition of "realism" per EPR (their "elements of reality"), then naturally you disagree about Bell.

Not too many will be standing with you, but there are always a determined few.

DrChinese: stevendaryl and his comment to rubi, quoted by me immediately above, leaves me wondering: Why are there not a lot more standing with me?

 

[DrChinese]

But realism under Bohr's insight (that a measurement may perturb the measured system) is not refuted.

The hidden variables could be part of the measurement apparatus in Bell's form. What you are saying is not supported in the literature, regardless of your allusions otherwise.

 

[DrChinese]

DrChinese, stevendaryl and his comment to rubi, quoted by me immediately above, leaves me wondering: Why are there not a lot more standing with me?

Because you have adopted definitions at odds with EPR. Naturally your conclusions might differ in that case, and in fact they have. The EPR realism definition is compelling to the typical reader, which is why the EPR paper is so important.

 

[stevendaryl]

As I wrote above: it would never occur to me to make such an assumption under EPRB! So I am inclined to say that this statement should be known as the 'strawman-realist' assumption.

Well, it's a coherent notion of realism, and if it's a strawman, I've never seen a non-strawman concept of realism.

For I am a realist under this article by d'Espagnat (which Bell endorsed): http://www.scientificamerican.com/media/pdf/197911_0158.pdf

2: On p.158 we see these principles of local realism: (i) realism -- regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers; (ii) locality -- no influence of any kind can propagate superluminally; (iii) induction -- legitimate conclusions can be drawn from consistent observations.​

Well, Bell's "strawman" notion of realism is an attempt to make sense of those properties. If you can come up with a model that is realistic in that sense but not in Bell's, I'd like to hear it.

 

[stevendaryl]

Note: This then leaves me wondering why the general claim associated with Bell's theorem is that "local-realism" is refuted? I can accept that local strawman-realism is refuted. But realism under Bohr's insight (that a measurement may perturb the measured system) is not refuted.

I would call that Heisenberg's insight. But Bell's theorem rules out such a notion of realism, as well.

You might think that a more general notion of realism is this:

Instead of saying that the measurement results are determined ahead of time, you can allow that the act of measurement is a nondeterministic process, so the outcome of the measurement depends on

1. Facts about the particle being measured.
2. Facts about the device doing the measurement.
3. Randomness in the measuring process.

However, the only way using realism to get perfect anti-correlations in EPR is if the randomness plays no role. If in the EPR experiment using spin-1/2 particles, Alice measures spin up along axis \vec{a}, then she knows with 100% certainty that Bob will measure spin down along axis \vec{a}. There is no randomness in Bob's result.

So if you want to try to use the idea that Alice's measurement disturbs the system, then you would have to allow for Alice's actions to disturb Bob's particle, which would imply faster-than-light influences.

So no, Bell's assumptions is not just a "straw man".

 

[edguy99]

... Instead of saying that the measurement results are determined ahead of time, you can allow that the act of measurement is a nondeterministic process, so the outcome of the measurement depends on

1. Facts about the particle being measured.
2. Facts about the device doing the measurement.
3. Randomness in the measuring process.

However, the only way using realism to get perfect anti-correlations in EPR is if the randomness plays no role...

Well said, easy to understand. Can you get this published in a peer reviewed journal so we can quote it?

 

[zonde]

Are we able to assume that realistic properties do exist independent of measurement, but that the act of measurement changes their distribution?

I would say that this is possible under condition that randomness in measurement process is restricted by non-local conservation of phase.

 

[Boing3000]

(i) realism -- regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers; (ii) locality -- no influence of any kind can propagate superluminally; (iii) induction -- legitimate conclusions can be drawn from consistent observations.

I see not difference between (i) and stevendaryl post #171.
Are you saying that (i) should not be called realism but strawman-realism ?

Note: This then leaves me wondering why the general claim associated with Bell's theorem is that "local-realism" is refuted?

Because any local phenomenon would require magic influence over observers, not FLT influence, but unrealistic influence which dependent only on what Bob and Alice are(not ?) doing locally.

 

[rubi]

We're talking in the context of Bell's inequality, which was derived under the assumption that we had a local realistic theory. With such a theory, the variables a_n, a_n', b_n, b_n' exist independently of whether anybody measures them, and so it makes sense to talk about averages of them.

You can certainly reject that assumption, which means rejecting the assumption that Bell proved false, anyway.

Well, I thought it was N88's point that the BI can't be proved without this additional assumption, so I pointed it out, but after his latest comments, I'm not so sure anymore, what his point is.

As I wrote above: it would never occur to me to make such an assumption under EPRB! So I am inclined to say that this statement should be known as the 'strawman-realist' assumption.

The word "realism" in the context of Bell's inequality refers to a well-defined technical assumption in the proof of Bell's inequality. It has nothing to do with the fuzzy concept of realism in philosophy. Of course, QM describes reality, because it is consistent with experiments. The violation of Bell's inequality just implies that reality is much more peculiar than classical physicists imagined it to be. In particular, this means that experimental results aren't pre-determined by hidden variables (if you are a Bohmian, you will expoloit the non-locality loophole instead). There are much more adequate ways to phrase the conclusion of Bell's theorem, such as: "No local hidden variable theory / no local classical theory / no local counterfactually definite theory can reproduce the predictions of QM." It is then absolutely clear that one is talking about a sharp mathematical criterion rather than some ill-defined philosophical concept. But of course, terminology in science is not fixed and thus you will still find the term "realism" in many publications. Nevertheless, what we can take away is that reality can't work as one would naively imagine, so Bell's theorem does indeed impact our conception of reality.

 

[DrChinese]

Well said, easy to understand. Can you get this published in a peer reviewed journal so we can quote it?

Generally accepted science does not require references. If someone posts without references and makes statements that go against standard physical theory, they are subject to challenge. So it is best to stick to the mainstream, and clearly identify areas in which your opinion does not clearly reflect that.

 

[Jilang]

Would this destroy realism?

1. Facts about what is being measured
2. Facts about the measuring equipment
3. Randomness of the orientation of what is being measured relative the measuring equipment.

 

[N88]

Well, I thought it was N88's point that the BI can't be proved without this additional assumption, so I pointed it out, but after his latest comments, I'm not so sure anymore, what his point is. Emphasis added.

This was my point. This remains my point. I appreciated that you appeared to be making the same point so clearly in your earlier posts.

I cannot see where my later comments depart from this point. For me, the well-defined "realism" in the context of Bell's theorem is most clearly evident when Bell (1964) moves from his (14a) to his (14b). If the lambdas are appropriately paired, (14a) is a testable and readily validated formulation under EPRB. But (14b) is false under EPRB because it equates to his (15) which is known to be false under EPRB.

This falsity must therefore arise from the "realist" assumption that Bell used in moving from valid (14a) to invalid (14b): by my analysis, that assumption is (as given in the OP with added details there re consequences).

[A(b,λ)]² = 1. (X)

Now, in that it has never occurred to me to make such an assumption (me knowing it to be false via QM,* without reference to any of Bell's work), this is "straw-man realism" for me; a false argument advanced and refuted.

However, no matter the terminology, Bell and I agree: (15) is false under EPRB. But nowhere do I see Bell's recognition that (X) above is false. (See Bell's (1990) dilemmas re AAD and locality

• • http://www.quantumphil.org./Bell-indeterminism-and-nonlocality.pdf

and this next.)

Now, in that (X) may not be a straw-man argument for others, I refer them to d'Espagnat's (1979) article (endorsed by Bell). There, in my opinion, they have defined "naive realism" in that (as I see it): they inferred the measured output was the same as the measurement input. A⁺ output leads to their inference "A⁺ is a particle property."

Thus, to me, (X) is licensed by them under "naive-realism" (or, for me, under "straw-man realism") both of which are false under EPRB.

I therefore maintain my conclusion that (using some of your terms): "BI cannot be proved without assumption (X)" which is a well-defined technical assumption in the derivation of Bell's inequality.

The word "realism" in the context of Bell's inequality refers to a well-defined technical assumption in the proof of Bell's inequality. It has nothing to do with the fuzzy concept of realism in philosophy. Of course, QM describes reality, because it is consistent with experiments. The violation of Bell's inequality just implies that reality is much more peculiar than classical physicists imagined it to be. In particular, this means that experimental results aren't pre-determined by hidden variables (if you are a Bohmian, you will exploit the non-locality loophole instead). There are much more adequate ways to phrase the conclusion of Bell's theorem, such as: "No local hidden variable theory / no local classical theory / no local counterfactually definite theory can reproduce the predictions of QM." It is then absolutely clear that one is talking about a sharp mathematical criterion rather than some ill-defined philosophical concept. But of course, terminology in science is not fixed and thus you will still find the term "realism" in many publications. Nevertheless, what we can take away is that reality can't work as one would naively imagine, so Bell's theorem does indeed impact our conception of reality.

In my terms (which, it seems to me, can also be yours): We can take away from Bell's work that "naive realism" does not work (as naive persons might naively imagine), so Bell's Theorem does indeed impact naive conceptions of reality. * (All in line with prior work by Heisenberg, Bohr, etc. And now you, rubi : QM describes reality, because it is consistent with experiments. The violation of Bell's inequality just implies that reality is [somewhat] more peculiar than classical physicists imagined it to be.

 

[DrChinese]

QM describes reality, because it is consistent with experiments. The violation of Bell's inequality just implies that reality is much more peculiar than classical physicists imagined it to be.

Finally you are saying something that makes sense. Of course, that is still saying that classical realism is untenable. Which is in fact what Bell's Theorem says is incompatible with the predictions of QM.

 

[N88]

Finally you are saying something that makes sense. Of course, that is still saying that classical realism is untenable. Which is in fact what Bell's Theorem says is incompatible with the predictions of QM.

Thanks DrC, with due acknowledgment to RUBI.

 

[N88]

I see not difference between (i) and stevendaryl post #171.
Are you saying that (i) should not be called realism but strawman-realism ?

Thanks for seeking clarification. No, I am not saying that at all. I readily and happily accept d'Espagnat's Bell-endorsed principles (i)-(iii).

Re other matters, please see my more detailed reply to rubi above.

 

[stevendaryl]

Now, in that (X) may not be a straw-man argument for others, I refer them to d'Espagnat's (1979) article (endorsed by Bell). There, in my opinion, they have defined "naive realism" in that (as I see it): they inferred the measured output was the same as the measurement input. A⁺ output leads to their inference "A⁺ is a particle property."

Well, to call it "naive realism" or "strawman realism" suggests that there is some "non-naive" notion realism under which QM might be a local realistic theory. But what is that?

The claim that the "measured output was the same as the measurement input"---I'm not exactly sure what you mean by that, but you mean the fact that claim that the measurement reveals a pre-existing hidden variable. If that's what you mean, that is not an assumption, that is a conclusion from the fact that QM predicts perfect correlations/anti-correlations in EPR-type experiments.

Local realism to me (I'm not sure if this is naive local realism, or not, but if it is, I would like to see what is non-naive realism) says that the result of a measurement depends on what the local situation is. To me (not everyone agrees with this), nondeterminism is compatible with local realism, so the outcome of a measurement in a local realistic theory could potentially be nondeterministic. But in a locally realistic theory, if the outcomes are probabilistic, then the probabilities of various outcomes can only depend on local facts.

So potentially, you could, for the anti-correlated EPR experiment have a locally realistic theory that would say:

• The probability that Alice measures spin-up for her particle is some function P_A(\alpha, \lambda, O_A, T_A), where \alpha is the setting of Alice's detector, \lambda is a variable describing the production of the twin pair, O_A is a variable describing other miscellaneous properties of Alice, her detector, the measurement process, etc., and T_A is a variable describing Alice's particle's travels from the point of creation to the point of detection.
• Similarly, the probability that Bob measures spin-up for his particle is some function P_B(\alpha, \lambda, O_B, T_B)

So you don't have to assume that the measurement results are set in stone from the beginning; they might potentially depend on all sorts of things. However, the perfect anti-correlation implies that if Alice measures spin-up at detector setting \alpha, then Bob certainly will not measure spin-up at that setting, and vice-versa. This implies that for a fixed \alpha and \lambda,

• P_A(\alpha, \lambda, O_A, T_A) P_B(\alpha, \lambda, O_B, T_B) = 0

One or the other probability must be zero, since they never both happen. Also, it never happens that they both measure spin-down at the same detector setting, either. Since empirically, you either get spin-up or spin-down, the probability of spin-down is 1 - the probability of spin-up. So we have:

• (1 - P_A(\alpha, \lambda, O_A, T_A))(1 - P_B(\alpha, \lambda, O_B, T_B)) = 0

Those two facts about probability tell us that for any \alpha and \lambda:

• Either P_A(\alpha, \lambda, O_A, T_A) = 0 or P_A(\alpha, \lambda, O_A, T_A) = 1
• Either P_B(\alpha, \lambda, O_B, T_B) = 0 or P_B(\alpha, \lambda, O_B, T_B) = 1

So even though we allowed the results to be probabilistic, the perfect anti-correlations imply that the result must be deterministic. For some values of \alpha and \lambda, it is certain that Alice will get spin-up and that Bob will get spin-down. For other values, it is certain that Bob will get spin-up and Alice will get spin-down.

So the conclusion that the result is deterministic follows from the assumption of perfect anti-correlation and the assumption that whatever probabilities are involved depend only on local variables.

 

[Boing3000]

We can take away from Bell's work that "naive realism" does not work (as naive persons might naively imagine), so Bell's Theorem does indeed impact naive conceptions of reality.

That is incorrect. By the definition of realism that you've finally agreed with (and everybody does because there is no other definition), a very naive person, helps by Occam's razor, will conclude that locality do not work. An that is trivial to realistically implement, as already shown.

Now, if you have another/better operational definition of realism, I would also appreciate that you explain it ... precisely.

 

[N88]

Summarising my current position:

1. In post #190 [ https://www.physicsforums.com/threa...nce-of-bells-math.904029/page-10#post-5710931 ] I have summarised my understanding of the answer to the OP: "What is the physical significance of Bell's math?"

2. I thank all who have helped me arrive at that position. (I now realize that my starting point -- "having rejected Bell's thesis from day-one" -- has contributed to many crossed and frustrating exchanges.) I look forward to further comments that might clarify, correct or improve that summary; etc.

3. As to unfinished business that goes beyond the OP, I think it best if new specifically-titled threads are opened.

4. I plan to reply here (soon) to some open matters. I need to be careful about non-mainstream ideas.

With my thanks again,

N88

 

[Zafa Pi]

A very relevant and easy to read paper by Guy Blaylock, "The EPR paradox, Bell’s inequality, and the question of locality" addresses a number questions that have come up in this thread.

What's hard for me to understand is what would count as a non-realistic theory.

Blaylock's answer is MWI

I hate it when people talk about counterfactual definiteness, because to me that sends people off onto a philosophical and meaningless discussion about whether counterfactual definiteness is a desirable property, or what it means, and whether nondeterministic theories are counterfactually definite. It's a mess that doesn't make any difference. It's a red herring.

It's just that I don't think it clarifies anything. A local, nondeterministic theory violates CFD, so violating CFD is not a big deal, it seems to me, and it doesn't do anything to understand the difference between a quantum theory and a classical theory.

Blaylock makes the opposite case.

I'm just saying that I think you're wrong. If nonlocality is defined in Bell's terms, then QM is either nonlocal, or one of the weird acausal interpretations (superdeterminism, back-in-time causality) must be true.

Again Blaylock would say MWI is local.

I would say no. I don't consider FTL and nonlocality to be synonymous.

You may be right, but what about Blaylock's page 6?

Of course, that is still saying that classical realism is untenable. Which is in fact what Bell's Theorem says is incompatible with the predictions of QM.

What about BM? Or is that not classical?

There are a couple places in Blaylock's paper I found flawed. For example, the way he deals with the uncertainty principle and says p13, "Thus, a definite momentum measurement forces the physical system into an indefinite superposition of distinct position eigenstates."
But for the most part I liked the paper.

 

[stevendaryl]

A very relevant and easy to read paper by Guy Blaylock, "The EPR paradox, Bell’s inequality, and the question of locality" addresses a number questions that have come up in this thread.

Blaylock's answer is MWIBlaylock makes the opposite case.

Well, I stand by what I said earlier. I don't think that CFD is an important element, and I don't think that lacking CFD is a very useful way to talk about MWI. To me, MWI is not a realistic model, in the sense that questions along the lines of: "Will Bob measure spin-up along axis \vec{a}?" don't have answers. EPR, in talking about "elements of reality" were making the assumption that Alice, by measuring spin-up along axis \vec{a} was learning something definite about Bob far away--that he will definitely measure spin-down along that axis. MWI abandons the assumption that Alice learns anything at all about Bob. There is no "fact of the matter" about what result Bob will get. It's not just failure of CFD, which as I said, is a feature of most stochastic models. It's not just that what Bob would have gotten if he had performed a different measurement is undefined---in MWI what he ACTUALLY got was undefined. He got spin-up and he got spin-down.

So I don't agree with Blaylock's way of putting it.

The other point of disagreement is that I don't consider MWI local. In MWI, the state of the universe is the wave function, and the wave function is not a local model. The wave function is a function on configuration, not on physical space, while a local model in the sense of Bell is one where there is a state of the universe that "factors" into states of local neighborhoods of the universe. That isn't the case with MWI.

MWI does not have FTL influences, but it isn't a local model in the sense of Bell.

 

[Zafa Pi]

To me, MWI is not a realistic model

And for Blaylock that implies nonCFD.

So I don't agree with Blaylock's way of putting it.

OK, to each his own.

 

[zonde]

A very relevant and easy to read paper by Guy Blaylock, "The EPR paradox, Bell’s inequality, and the question of locality" addresses a number questions that have come up in this thread.

There is interesting statement toward the end of the paper:

The many-worlds interpretation is not only counterfactually indefinite, it is factually indefinite as well.

So we might ask if according to author it is counterfactual definiteness that is questioned or is it rather factual definiteness that should be given up to keep locality.