Bell's equalities

[minkwe]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

Some times, the terms needed to understand the meaning of the inequality are precisely the ones that have been hidden/removed. So in this thread, I'm hoping we would carry out a simple exercise with Bell's inequality. For Bell's theorem, using the wikipedia form:


In other words using my earlier symbols:





What is ?

Let us complete the expression with the full form of z. Let us do the derivation of the equality rather than the inequality. Once we are done, let us examine what it means to say .

[FrediFizzx]

This seems a bit abstract. I suppose h is an index for hidden variable? I think you explained the inequalities better when describing the indices that are usually missing.
.

[minkwe]

This seems a bit abstract. I suppose h is an index for hidden variable? I think you explained the inequalities better when describing the indices that are usually missing.
.

I'm just using the form of Bell's inequality from Wikipedia but any version would do. I'm more interested in completing the expression than in the indices at this point

[Heinera]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

I guess you meant to write .

But I anyway don't see why must contain any extra information here. If I say

If I throw a dice with outcome , then

I could write it as

If I throw a dice with outcome , then .

But I don't see how this would contain any extra relevant information, except for the trivial fact that it is the difference between and 6.

[minkwe]

I guess you meant to write .

Thanks, it was a flipped sign. I've corrected it.

But I anyway don't see why must contain any extra information here.

Let us complete it first before deciding whether it contains additional information. Once we have the full expression, then you are welcome to argue that it contains no additional information if you want.

[FrediFizzx]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

Some times, the terms needed to understand the meaning of the inequality are precisely the ones that have been hidden/removed. So in this thread, I'm hoping we would carry out a simple exercise with Bell's inequality. For Bell's theorem, using the wikipedia form:


In other words using my earlier symbols:





What is ?

Let us complete the expression with the full form of z. Let us do the derivation of the equality rather than the inequality. Once we are done, let us examine what it means to say .

So . Now what?
.

[minkwe]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

Some times, the terms needed to understand the meaning of the inequality are precisely the ones that have been hidden/removed. So in this thread, I'm hoping we would carry out a simple exercise with Bell's inequality. For Bell's theorem, using the wikipedia form:


In other words using my earlier symbols:





What is ?

Let us complete the expression with the full form of z. Let us do the derivation of the equality rather than the inequality. Once we are done, let us examine what it means to say .

So . Now what?
.

It's not so easy, I'm not asking you to start from the inequality and work backwards, I'm asking that we follow the derivation step by step but keeping all terms in the equality instead of just summarising it as an inequality. It is probably easier to start with the Probabilistic version of Bell's inequality using a venn-like diagram, since it is obvious there to see where the "greater than or equal to sign originates". I'll post something soon if nobody else does.

[minkwe]

Okay, here is Bell's derivation, step by step:

1.
2.
3.
4.
5.
6.
So far, we have just reproduced Bell's steps from his original derivation. This is the point at which the inequality is introduced. But let us carry on his logic and try not to introduce any inequality, keeping the equality throughout to the end to see what remains when we construct the P(b,c) term.

[gill1109]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

I was taught that is *defined* to mean that .

So what you call "an incomplete form" is not *incomplete* at all. Of course, it can be useful to know how the various symbols you are using were originally defined.

Usually, we don't just introduce new symbols but we also prove useful properties of them, so that we expand not just the collectioon of words in our language but also establish new ways to derive true statements in the language from already established true statements.

For instance it is a *theorem* that if and then

[minkwe]

On these forums, we have spent many years arguing about inequalities. But do we really understand what they mean? I'm becoming convinced that often, inequalities are easy to misinterpret because they are by definition incomplete mathematical expressions. For example the expression

can be written in incomplete form as

But this obscures the claim about

I was taught that is *defined* to mean that .

So what you call "an incomplete form" is not *incomplete* at all. Of course, it can be useful to know how the various symbols you are using were originally defined.

Usually, we don't just introduce new symbols but we also prove useful properties of them, so that we expand not just the collection of words in our language but also establish new ways to derive true statements in the language from already established true statements.

For instance it is a *theorem* that if and then

I think you missed the point, but I'll find a better way of making it. But in the mean time, I will make a related point. We can continue from equation 6 above, using the same logic which Bell used:

6.
7.

since 1/A = A we just factored out in the same way as we factored out to go from 5 to 6.

taking absolute values, we end up with the following

8.
9.

The interesting thing is that this inequality can be tested with just two experiments with all measurements, on the same particle pairs. No need to perform a third experiment since all the terms are available. And since it was derived by using exactly the same assumptions, any process which legitimately "violates" the other inequality should also violate this one.

Any objections to this analysis?

[Heinera]

Any objections to this analysis?

Yes. How do you get from (8) to (9)?

[minkwe]

Sorry there was an error, it should be the following:


7.

since 1/A = A we just factored out in the same way as we factored out to go from 5 to 6.
taking absolute values, we end up with the following

8.
9.

You can compare with equations 14 to 15 of Bell's original paper. The only difference being we have factored out in addition to .

The interesting thing is that this inequality can be tested with just two experiments with all measurements, on the same particle pairs. No need to perform a third experiment since all the terms are available. And since it was derived by using exactly the same assumptions, any process which legitimately "violates" the other inequality should also violate this one.

Any objections to this analysis?

[Heinera]

This is still wrong. How do you get from (7) to (8)?

(In details, please.)

[minkwe]

7.
7a.
7b.
8'.

Okay, I give up, obviously do not have the time to focus on this. 8 should be absolute values not square brackets therefore 9 does not follow from 8 but this is irrelevant to the point I wanted to make. QM should easily violate 8'. Right?

Nevermind.

[Heinera]

8' is correct. But it's also pretty uninteresting, since it can't be tested empirically, not even in a thought experiment. Nor does it have any QM counterpart that you could compare with. The only definite thing you can theoretically conclude about the RHS in 8' is that it can never exceed 2.

[minkwe]

8' is correct. But it's also pretty uninteresting, since it can't be tested empirically, not even in a thought experiment. Nor does it have any QM counterpart that you could compare with. The only definite thing you can theoretically conclude about the RHS in 8' is that it can never exceed 2.

Experimentally, as N>>1, 8' becomes:



Please explain why this can't be measured experimentally but Bell's version can:



or

[Heinera]

Experimentally, as N>>1, 8' becomes:

This makes no sense. Your LHS is not . And the RHS has nothing to do with .
What is ? Is it supposed to be ?

[minkwe]

Experimentally, as N>>1, 8' becomes:

This makes no sense. Your LHS is not . And the RHS has nothing to do with .
What is ? Is it supposed to be ?

is an index variable for an iteration in an experimental. It makes a lot of sense, what are you talking about?

for large N, 8' is equivalent to

or in short, in the limit as :


Or even shorter, let , then you get


I haven't done anything here that is different from the same logic that Bell applied, and which is applied to compare Bell's inequalities to experimental data.
Please explain your issue with the expression. The absence of lambda is irrelevant to the equivalence. These are empirical quantities, and you can relabel as if you please.

[Heinera]

For one thing, why are there no function B(...) on LHS? Have you completely forgotten about Bob?

[minkwe]

For one thing, why are there no function B(...) on LHS? Have you completely forgotten about Bob?

Please take a look at Bell's paper again. Page 406.