To understand Bell's inequality

[mananvp]

TL;DR Summary

Hello, I have read and watch some informative videos on bell's inequality. I think I understood it but as I go deep more questions arises.

First I give detail of what I think I understood so far.

Suppose, there are three angles A, B, C separated by 120° angles. A can measure + (spin up in A direction, we call it A+), and - (spin down in A direction, we call it A-). Same goes for B+, B- and C+, C-.

I have choose A direction to measure +, B to measure + and C to measure -. So this way my equation looks like below.

P(A+, B+) ≤ P(A+, C-) + P(C-, B+)

I could find that this stays true if hidden variables theory are correct and it violates when quantum mechanics is correct using probability as below.

P(X, Y) = cos²(θ_(xy)/2)

Is my understanding correct so far? Specially the equation I wrote is correct? I need to understand more but I want to confirm that from where I started is correct or not.

Thanks.

 

[PeroK]

The point is that we are talking about two measurements of spin/polarization, each on one particle in an entangled pair. So that: $$P(A+, B+)$$ is the probability that the first particle is measured in the A direction and the result is +; and, the second particle is measured in the B direction and the result is +.

Does that make sense?

 

[mananvp]

Yes. I am also following the same analogy.

Like if there are two observer Alice and Bob then if Alice measure spin + in A direction and Bob measure spin + in B direction for the same particles pair. I write it as below

P(A+_{(measured by Alice)}, B+_{(measured by Bob)})

Thanks for replying

 

[PeroK]

Yes. I am also following the same analogy.

Like if there are two observer Alice and Bob then if Alice measure spin + in A direction and Bob measure spin + in B direction for the same particles pair. I write it as below

P(A+_{(measured by Alice)}, B+_{(measured by Bob)})

Thanks for replying

Okay so far, then!

 

[mananvp]

Ok. so, my understanding is correct so far then I have a question.

I have seen an another analogy like when Alice and Bob meet one another and they compare results, if they found different result more than same result then the hidden variable theory is correct.

P(different result) > P(same result)

If the above equation do not work we can tell that hidden variables theory is not correct.

I can understand the analogy too. I can also understand that if do experiment we get

P(different result) = P(same result)

thus hidden variables theory is not correct.

But, I can't understand from where the analogy of above inequality comes from? How can I derive above equation from below one?

P(A+, B+) ≤ P(A+, C-) + P(C-, B+)

Thanks.

 

[PeroK]

But, I can't understand from where the analogy of above inequality comes from? How can I derive above equation from below one?

P(A+, B+) ≤ P(A+, C-) + P(C-, B+)

Thanks.

This is the key equation. You must do the experiment many times to get a statistically reliable estimate of the various probabilities. Sometimes Alice measures in the A direction and Bob measures in the B direction; sometimes Alice measures in the A direction and Bob in the C direction; and sometimes Alice measures in the C direction and Bob in the B direction. This gives us an experimental estimate of the above probabilities.

If we have hidden variables, then the above inequality must hold.

If, however, we make a certain choice of angles between A, B and C, then the equality fails experimentally.

QM, on the other hand, predicts different probabilities, which do hold experimentally.

 

[mananvp]

PeroK said:
-----------------------------------
On the other hand, if we make a certain choice of angles between A, B and C, then the equality fails experimentally.

QM, on the other hand, predicts different probabilities, which do hold experimentally.
-----------------------------------
Can you please elaborate above statements little bit?

And yeah that was key equation but for specific angles. The general explanation is followed by below one

P(different result) > P(same result)

Because both observer do not know what angle to be chosen during experiments. Can you please give me some insight or idea how to generalize that equation?

Thanks

 

[PeroK]

If we make a certain choice of angles between A, B and C, then the equality fails experimentally.

Because both observer do not know what angle to be chosen during experiments. Can you please give me some insight or idea how to generalize that equation?

Thanks

There's no more maths after that equation. It's a question of doing the experimental thousands of times to see whether the inequality holds.

Both observers agree in advance what angles to choose. They can randomise each individual measurement, but you can't just choose any old angle every time: you must repeat the same choice of angles thousands of times.

 

[mananvp]

There's no more maths after that equation. It's a question of doing the experimental thousands of times to see whether the inequality holds.

Both observers agree in advance what angles to choose. They can randomise each individual measurement, but you can't just choose any old angle every time: you must repeat the same choice of angles thousands of times.

Sorry to say but I don't get the idea about how choosing different different angles choosing thousands of time come to conclusion that

$$P(S_{ame~Result}) \lt P(D_{iff~Result})$$

My curiosity is not ending here. After experiments lot of times we can check for each variant of equation.
Let me show my unsuccessful effort of looking into this. Here, are each variant which needs to be check for inequality.

$P(A+,~B+) \leq P(A+,~C-) + P(C-,~B+)$
$P(A+,~C+) \leq P(A+,~B-) + P(B-,~C+)$
$P(A+,~A+) = 0$

$P(B+,~A+) \leq P(B+,~C-) + P(C-,~A+)$
$P(B+,~C+) \leq P(B+,~A-) + P(A-,~C+)$
$P(B+,~B+) = 0$

$P(C+,~A+) \leq P(C+,~B-) + P(B-,~A+)$
$P(C+,~B+) \leq P(C+,~A-) + P(A-,~B+)$
$P(C+,~C+) = 0$

First two inequality from each set violates in experiments. Here insight is left hand side of the equation are all probabilities for same result. The 9 combinations are possible to find combined probability of obtaining same results.

$$P(S_{same}) = \frac 1 9 [(P(A+,~B+) + P(A+,~C+) + P(B+,~A+) + P(B+,~C+) + P(C+,~A+) + P(C+,~B+)]$$

We are ignored putting 0 probability term in above one.

Now, the same we can do if we need the probability of different results.

$$P(D'_{iff}) = \frac 1 9 [(P(A+,~B-) + P(A+,~C-) + P(B+,~A-) + P(B+,~C-) + P(C+,~A-) + P(C+,~B-)]$$ (I chose D' here for a reason)

We can see that all the terms appear here, we can find those terms in right hand side of main equations. So, we can write this like

$$P(S_{same}) \leq P(D'_{iff}) + \frac 1 9 [P(C-,~B+) + P(B-,~C+) + P(C-,~A+) + P(A-,~C+) + P(B-,~A+) + P(A-,~B+)]$$

I still haven't introduce $P(A+, A-) = 1/9$, $P(B+, B-) = 1/9$, $P(C+, =C-) = 1/9$ yet.

If I add the three terms in $P(D'_{iff})$. I get

$$P(D_{iff}) = P(D'_{iff}) + 1/3$$

Now, I have added 1/3 to right hand side so I can conclude that the equation becomes $\lt$ instead of $\leq$.

$$P(S_{same}) \lt P(D_{iff}) + \frac 1 9 [P(C-,~B+) + P(B-,~C+) + P(C-,~A+) + P(A-,~C+) + P(B-,~A+) + P(A-,~B+)]$$

The extra term in right hand side $\frac 1 9 [P(C-,~B+) + P(B-,~C+) + P(C-,~A+) + P(A-,~C+) + P(B-,~A+) + P(A-,~B+)]$ is I think should not be part of the equation because it is not in our previous set of different result, so if I remove it from right hand side I can't be sure that equation still stay $\lt$. It might become $\leq$ or it might become $\gt$.

Here I am stuck and do not find a way to come up for

$$P(S_{same}) \lt P(D_{iff})$$

Please point me out if I have done some mistakes during my calculation.
Thanks.

 

[PeroK]

Here I am stuck and do not find a way to come up for

$$P(S_{same}) \lt P(D_{iff})$$

Please point me out if I have done some mistakes during my calculation.
Thanks.

I don't know what your equation has to do with Bell's theorem. There are no further calculations beyond:

P(A+, B+) ≤ P(A+, C-) + P(C-, B+)

That is Bell's Theorem. That's the equation to be experimentally tested.

 

[mananvp]

I don't know what your equation has to do with Bell's theorem. There are no further calculations beyond:

P(A+, B+) ≤ P(A+, C-) + P(C-, B+)

That is Bell's Theorem. That's the equation to be experimentally tested.

Ok. It was my bad. I was trying to understand other way around. Now, I started from 8 possibility of hidden variables configurations for 3 directions and 9 possible directions to be chosen by both observers and I got 72 states and I could find that 5/9 was different result and 4/9 was same result.

If I take three special states $P(A+,~B+),~P(A+,~C-),~P(C-,~B+)$ from that 72 states and generalize probability of occurrence all 8 configurations, I still could confirm the inequality.

Thanks for discussing with me.

 

[andrew s 1905]

This is the key equation. You must do the experiment many times to get a statistically reliable estimate of the various probabilities. Sometimes Alice measures in the A direction and Bob measures in the B direction; sometimes Alice measures in the A direction and Bob in the C direction; and sometimes Alice measures in the C direction and Bob in the B direction. This gives us an experimental estimate of the above probabilities.

If we have hidden variables, then the above inequality must hold.

If, however, we make a certain choice of angles between A, B and C, then the equality fails experimentally.

QM, on the other hand, predicts different probabilities, which do hold experimentally.

Can I ask if the need for many trials is due to experimental limitations or that the correlations predicted by QFT only hold statistically and not for each specific measurement?For example, in a two spin state entanglement experiment if Bob and Alice both measure in the same direction will they definitely (in a perfect experiment) always get an Up Down pair or only on average?
Regards Andrew

 

[PeroK]

Can I ask if the need for many trials is due to experimental limitations or that the correlations predicted by QFT only hold statistically and not for each specific measurement?For example, in a two spin state entanglement experiment if Bob and Alice both measure in the same direction will they definitely (in a perfect experiment) always get an Up Down pair or only on average?
Regards Andrew

The whole thing is statistical. If Alice and Bob measure in the same direction, then the anti-correlation is perfect, and you can't tell the difference between QM and the obvious hidden variables theory.

The trick is for Alice and Bob to make measurements at different angles, where the correlations are not perfect; and, moreover, to have a well-chosen set of angles which makes it impossible for hidden variables to match QM.

This was Bell's great insight. That if you mixed and matched the angles you had an inequality for hidden variables that wouldn't apply to QM. Your experiment must move away from the simple cases where Alice and Bob always get opposite results.

Any probabilistic result needs a large number of tests to achieve statistical confidence.

 

[DrChinese]

Can I ask if the need for many trials is due to experimental limitations or that the correlations predicted by QFT only hold statistically and not for each specific measurement?For example, in a two spin state entanglement experiment if Bob and Alice both measure in the same direction will they definitely (in a perfect experiment) always get an Up Down pair or only on average?
Regards Andrew

As PeroK said: " The whole thing is statistical. If Alice and Bob measure in the same direction, then the anti-correlation is perfect, and you can't tell the difference between QM and the obvious hidden variables theory. "

I.e. this is a special case, and there are several of those. Perfect correlations/anti-correlations are such. Also, there are cases where the match rate is 50-50 (random). At other angles, it is possible to see the issues involved a la Bell.

 

[stevendaryl]

I don’t remember the details, but I seem to remember some thought experiment that is more stark than EPR. Some experiment where a single measurement can distinguish between the predictions of quantum mechanics and the predictions of a local hidden variables theory. I will see if I can find it by Googling...

 

[stevendaryl]

What I am thinking of is the GHZ paradox...