Is BI really required to validate the Hidden Variables?

[Adel Makram]

By the definition of hidden variable (HVs); they are the parameters that determine what quantum states of two entangled particles will be after they are measured at particular measurement setups. Therefore and by definition, HVs should be reproducible and by undertaking measurements once or at any replica of times, the HVs could be estimated.
For example, if u1 and u2 are two possible states of entangled particles, and a1 and a2 are two chosen measurements setups for measuring particle 1 and 2 respectively, then running the experiment once (or many times to avoid error) will let us know the HVs at this particular setups. So if the outcomes of the measurements come against what quantum mechanics predicts for certain combination of a1 and a1, then hidden variable theory is true. No need to have three measurements parameters like what are used in Bell`s inequality to validate the existence of hidden variables.

 

[Strilanc]

First, Bell inequalities and Bell tests aren't used to confirm the existence of hidden variables. Instead, they exclude hidden variables. Bell tests show that a particular class of hidden variable theories is insufficient for explaining what we measure in experiments.

Second, your reasoning is backwards. The structure of your argument is akin to saying something like... "If it rains, the sidewalk will be wet. And the sidewalk is wet. Therefore it is raining.". But that's a bad argument, because there are other reasons the sidewalk could be wet. Analogously, there could be reasons besides "hidden variable theory is true" for "the outcomes of measurements come against what quantum mechanics predicts". For example, quantum mechanics could give the wrong predictions because reality is even weirder than quantum mechanics suggests, with the even weirder true theory still having no local hidden variables.

 

[DrChinese]

By the definition of hidden variable (HVs); they are the parameters that determine what quantum states of two entangled particles will be after they are measured at particular measurement setups. Therefore and by definition, HVs should be reproducible and by undertaking measurements once or at any replica of times, the HVs could be estimated.
For example, if u1 and u2 are two possible states of entangled particles, and a1 and a2 are two chosen measurements setups for measuring particle 1 and 2 respectively, then running the experiment once (or many times to avoid error) will let us know the HVs at this particular setups. So if the outcomes of the measurements come against what quantum mechanics predicts for certain combination of a1 and a1, then hidden variable theory is true. No need to have three measurements parameters like what are used in Bell`s inequality to validate the existence of hidden variables.

Since you can measure them, they are not hidden. Once you postulate hidden variables and say what attributes they might have, then you can begin putting together an inequality.

 

[Adel Makram]

Second, your reasoning is backwards. The structure of your argument is akin to saying something like... "If it rains, the sidewalk will be wet. And the sidewalk is wet. Therefore it is raining.". But that's a bad argument, because there are other reasons the sidewalk could be wet.

But wetting the sidewalk is a good negative result, because having a dry sidewalk is a good indicator of a non-raining status. Similarly, having a probabilistic outcome at certain combination of measurement setups is a good indicator that hidden variable theory can not be true because they must have a reproducible outcomes all times.

 

[Nugatory]

No need to have three measurements parameters like what are used in Bell`s inequality to validate the existence of hidden variables.

You may be misunderstanding the thought process here. Bell's inequality doesn't "validate" local hidden variable theories, it describes a prediction that all local hidden variable theories must make when applied to systems with three or more measurement parameters.

We then run experiments that see if that prediction is correct, and if when it's not, we have evidence (at this point, after decades of experimentation, the evidence is strong enough that we can reasonably call it "proof") that no local hidden variable theory can be correct.

 

[Adel Makram]

Since you can measure them, they are not hidden. Once you postulate hidden variables and say what attributes they might have, then you can begin putting together an inequality.

Why not? suppose we are living in a world where hidden variable theory is valid. Then measuring status of entangled particles at a given measurement setup will allow us to know what is the hidden variable at that particular setup. And because we know them, they are no longer hidden. The term hidden variable is not actually a scientific term.

 

[DrChinese]

The term hidden variable is not actually a scientific term.

In terms of this debate, hidden variable means the value of a observable that is not actually observed. It is hidden. The term "counterfactual" is also used. The value of an observable IF it had been observed rather than what actually was observed.

So to be concrete: An electron has up spin in the x-axis. What are its y-axis and z-axis spins? In QM, when you measure x, the y and z spins become indeterminate. In hidden variable theories, those have specific values. Do you follow that? Note there are 3 observables in this case.

 

[Adel Makram]

You may be misunderstanding the thought process here. Bell's inequality doesn't "validate" local hidden variable theories, it describes a prediction that all local hidden variable theories must make when applied to systems with three or more measurement parameters.

We then run experiments that see if that prediction is correct, and if when it's not, we have evidence (at this point, after decades of experimentation, the evidence is strong enough that we can reasonably call it "proof") that no local hidden variable theory can be correct.

I think that was just a linguistic problem of mine but it dose still conform with what I meant.

 

[Adel Makram]

In terms of this debate, hidden variable means the value of a observable that is not actually observed. It is hidden. The term "counterfactual" is also used. The value of an observable IF it had been observed rather than what actually was observed.

So to be concrete: An electron has up spin in the x-axis. What are its y-axis and z-axis spins? In QM, when you measure x, the y and z spins become indeterminate. In hidden variable theories, those have specific values. Do you follow that? Note there are 3 observables in this case.

I would just like to add "in hidden variable theories, those have specific values at any chosen angle of Stern-Gerlach device". Right?

 

[DrChinese]

I would just like to add "in hidden variable theories, those have specific values at any chosen angle of Stern-Gerlach device". Right?

Emphasis on "any", and also add: simultaneous. So you can measure 1 such angle, and you can deduce the value of a second (by measuring its entangled partner). That's 2, neither of which are hidden. The third is "counterfactual" or "hidden". It takes 3 for a Bell-type inequality.

 

[Nugatory]

Similarly, having a probabilistic outcome at certain combination of measurement setups is a good indicator that hidden variable theory can not be true because they must have a reproducible outcomes all times.

That conclusion does not follow. The counterexample that was used, for centuries before QM ever came along, is the toss of a six-sided die. The outcome is probabilistic (one chance in six of getting any of the six possible results) in essentially all measurement setups, yet the situation is completely described by a perfectly good hidden variable theory: the hidden variables are the initial position and momentum of each particle making up the die and the surface it lands on.

 

[Adel Makram]

That conclusion does not follow. The counterexample that was used, for centuries before QM ever came along, is the toss of a six-sided die. The outcome is probabilistic (one chance in six of getting any of the six possible results) in essentially all measurement setups, yet the situation is completely described by a perfectly good hidden variable theory: the hidden variables are the initial position and momentum of each particle making up the die and the surface it lands on.

Great, so if we conduct an experiment with perfect adjustment of the initial position and momentum of each particle making up the die, then the result of die tossing will be exactly reproducible, correct!. But what if the result comes to be probabilistic despite that adjustment. Then we must know that hidden variable theory is not the explaining theory. Here, no inequality is required, it is just be repeating the experiment and see if results are reproducible.

 

[Nugatory]

Since you can measure them, they are not hidden

It's something of a digression here, but it has always seemed to me that "hidden" is a bit of a red herring.

The thrust of the EPR program was that quantum mechanics was incomplete, that there was an underlying theory and that the probabilistic predictions of QM would emerge from the behavior of the dynamical variables of this underlying theory. A theory in which the variables were out there in plain sight for all to see would have suited this purpose just as well as one in which the variables are hidden (and is just as precluded by Bell's inequality).

 

[Nugatory]

Great, so if we conduct an experiment with perfect adjustment of the initial position and momentum of each particle making up the die, then the result of die tossing will be exactly reproducible, correct!. But what if the result comes to be probabilistic despite that adjustment. Then we must know that hidden variable theory is not the explaining theory.

However, the experiment with "perfect adjustment of the initial position and momentum of each particle" isn't the experiment that we're performing when we throw the die and isn't feasible. The experiments that we are performing with the die, in which we lack that perfect adjustment, neither confirm nor refute the proposition that the behavior of the die is governed by a hidden variable theory.

In quantum mechanical experiments, the experimental problem is even more serious as we don't have a candidate hidden variable theory (EPR didn't propose one, they just said that it would be a good idea to be looking for one) so we don't know what it is that we need to be perfectly adjusting.

Bell's genius is that he found a way of experimentally distinguishing quantum predictions from hidden-variable predictions without having to precisely specify what the hidden variable were, let alone their initial states.

 

[DrChinese]

Great, so if we conduct an experiment with perfect adjustment of the initial position and momentum of each particle making up the die, then the result of die tossing will be exactly reproducible, correct!. ...

You say so, but this was not clear centuries ago. The concept of hidden variable theories is that we have not yet discovered the source of apparent randomness in quantum observables. That QM was not complete, because it does not specify outcomes in individual trials.

You might ask if the outcome of a quantum measurement might rule out realistic (hidden variable) theories without invoking a Bell Inequality. The answer to that is yes; the GHZ test is such. This came after Bell, though.

 

[Adel Makram]

The thrust of the EPR program was that quantum mechanics was incomplete, that there was an underlying theory and that the probabilistic predictions of QM would emerge from the behavior of the dynamical variables of this underlying theory. A theory in which the variables were out there in plain sight for all to see would have suited this purpose just as well as one in which the variables are hidden (and is just as precluded by Bell's inequality).

In Bell`s type experiment and following EPR, particles must have well-defined spins on three directions.
The first particle of type (a,−b,c), for example, if measured along angle(a) would give spin-up, if measured along angle(b) would give spin-down and if measured along angle(c) would give spin-up. Its entangled particle would be of type (-a,b,-c).

If hidden variable theory is true, the probability of having spin-up along a-direction (a) and spin-up along b-direction (b) is equal to the sum of pairs that have the properties {(a,−b,c)&(-a,b,-c)} and {(a,−b,-c)&(-a,b,c)} divided by all numbers of entangled pairs. But according to QM, that probability is a function of the difference between two angles. Now removing c from the picture, would not affect the conclusion.

 

[Adel Makram]

I think I got it now. The idea of hidden variables is to reproduce what QM predicts but in a planed fashion. If we use two angles only, the prediction of QM may be also reproduced if hidden variables are there because there is no way to highlight the difference between them. It is only when we have three angles, which makes the whole difference and this must be in a form of inequality.

 

[Nugatory]

I think I got it now. The idea of hidden variables is to reproduce what QM predicts but in a planed fashion. If we use two angles only, the prediction of QM may be also reproduced if hidden variables are there because there is no way to highlight the difference between them. It is only when we have three angles, which makes the whole difference and this must be in a form of inequality.

You got it...