Hidden Variables and Quantum Mechanics

[LikesIntuition]

I'm reading up on interpretations of quantum theory, and I just came across Bell's theorem, which is confusing me. My main concern is this:

Why would quantum mechanics predict something different than hidden variables?

I hope that question is coherent enough. I'm not sure if I'm using the correct lingo. But basically I'm not seeing why "spooky action at a distance" would allow for different outcomes than local communication between systems, or just a situation where the values of aspects of the systems are defined, and we just can't know them.

Thanks in advance, and I'll be happy to try and explain my confusion better if necessary.

 

[atyy]

At least for non-relativistic quantum mechanics, quantum mechanics does not predict something different from hidden variables.

Quantum mechanics does predict something different from local hidden variables.

 

[Water nosfim]

It can't be local and can't be less then C , so for some of us it hidden , you shold look for back in time communication

 

[LikesIntuition]

Ok, so how would quantum mechanics predict something different from local hidden variables? Does it have to do with when the collapse of the vector state happens?

 

[atyy]

Ok, so how would quantum mechanics predict something different from local hidden variables? Does it have to do with when the collapse of the vector state happens?

In the usual demonstration that quantum mechanics violates the Bell inequalities, wave function collapse is not needed. The parts of QM that are needed to are non-commuting observables and entanglement.

The CHSH inequality, which is one type of Bell inequality, is presented in sections 4.1 and 4.2 of Scarani's lecture notes http://arxiv.org/abs/0910.4222v1. Those sections contain an explanation of what is meant by local hidden variables, and why violation of the inequality is indicates that local hidden variables are not possible, and a calculation that shows that quantum mechanics violates the inequality.

 

[DrChinese]

Why would quantum mechanics predict something different than hidden variables?

I hope that question is coherent enough. I'm not sure if I'm using the correct lingo. But basically I'm not seeing why "spooky action at a distance" would allow for different outcomes than local communication between systems, or just a situation where the values of aspects of the systems are defined, and we just can't know them.

QM does not, in and of itself, predict something "other than" hidden variables. I would say it is silent on the point. Deductions (via no-go theorems such as Bell) strongly imply that hidden variables don't exist or have extremely unusual properties ("grossly non-local").

Experimental evidence is quite strong too (though not absolute). Despite best efforts, no one has been able to discover a root cause to random outcomes of any quantum observations on non-commuting bases.

Finally, the idea that an observation is merely the updating of local information in an otherwise realistic world has been generally discredited.

 

[Water nosfim]

The random outcome come from part of the many worlds and the communication come from back in time allso from the many worlds , this way its statistic

 

[.Scott]

Ok, so how would quantum mechanics predict something different from local hidden variables? Does it have to do with when the collapse of the vector state happens?

The Rules of Quantum Mechanics have been developed from observations. It explains what happens - not why - and without regard for whether it can be readily envisioned or explained.

So when two entangled particles are measured, the results are coordinated as if the particles knew what combination of measurements were going to be made before either particle reached the measuring apparatus. When the experiment is really run, that is what really happens.

The QM rules that predict this were developed based on other experiments. One of the "real physicists" on this forum can check my assertion of what that rule is:

The rule applies to two entangled particles that have separated and are moving away from each other. The specific rule is that when the spins of these two particles are measured and the axis of measurement is the same, the particles will always appear to have opposite spins. When the measurements are taken on axis 180 degrees apart, they will always agree. For other angular differences, compute the square of the cosine of the angular measurement difference. This will give you the portion of the time when the measurements will agree or disagree - depending on whether they are closer to 180 degrees or 0 degrees apart. The rest of the time, the measurements will be independent.

Anyway, when that rule is applied to the experiment that Bell described, the results predicted by QM are shown (by Bell) to be inconsistent with any explanation based on local hidden variables. The results are very sensitive to being able to successfully measure a high percentage of the particles - since local hidden variables can explain the QM results if you allow those hidden value to affect the measurability of the particle. However, this experiment has been performed with ions where the measurement rates were about 99%.

 

[LikesIntuition]

Anyway, when that rule is applied to the experiment that Bell described, the results predicted by QM are shown (by Bell) to be inconsistent with any explanation based on local hidden variables. The results are very sensitive to being able to successfully measure a high percentage of the particles - since local hidden variables can explain the QM results if you allow those hidden value to affect the measurability of the particle. However, this experiment has been performed with ions where the measurement rates were about 99%.

Alright, this is where I'm confused. In what way would local hidden variables allow for different outcomes than QM? Don't the entangled systems need to "agree" with each other in either case? If we let a particle break down into two photons that travel opposite directions, for example, how would local hidden variables cause the photons to make different "choices" than QM would? Because in both cases we just end up using conservation laws to make the photons agree, right?

 

[DrChinese]

Alright, this is where I'm confused. In what way would local hidden variables allow for different outcomes than QM? Don't the entangled systems need to "agree" with each other in either case? If we let a particle break down into two photons that travel opposite directions, for example, how would local hidden variables cause the photons to make different "choices" than QM would? Because in both cases we just end up using conservation laws to make the photons agree, right?

Nope! That is where Bell comes into play. Your analogy works as long as certain angle settings (notably the same two) are used. But it falls apart when other angle settings are used (such as 120 degrees apart). You cannot construct a data set - even one hand picked) in which the QM predictions would be matched using the logic you adhere to.

This is why Bell is so important. You can easily see for yourself. All you need is about 10 or 20 examples that you make up yourself. You will find that (using your example) you cannot construct a result set that matches the QM prediction of a 75% match rate for 120 degrees difference UNLESS you know in advance WHICH angles you are measuring. For example, consider these 3 pairs of angles:

0, 120
120, 240
240, 0

To match QM and experiment, you must know which of the above are to be measured in each trial. This is called "observer dependent" reality. In EPR terms: reality "here" is a function of the nature of a measurement "there" (i.e. spooky action at a distance).

If you don't know which of the 3 pairs are to be measured, your best rate is 66.6% which is simply different than experiment and therefore cannot be correct. That is for Type II entangled photon pairs.

 

[LikesIntuition]

This is why Bell is so important. You can easily see for yourself. All you need is about 10 or 20 examples that you make up yourself. You will find that (using your example) you cannot construct a result set that matches the QM prediction of a 75% match rate for 120 degrees difference UNLESS you know in advance WHICH angles you are measuring. For example, consider these 3 pairs of angles:

0, 120
120, 240
240, 0

To match QM and experiment, you must know which of the above are to be measured in each trial. This is called "observer dependent" reality. In EPR terms: reality "here" is a function of the nature of a measurement "there" (i.e. spooky action at a distance).

If you don't know which of the 3 pairs are to be measured, your best rate is 66.6% which is simply different than experiment and therefore cannot be correct. That is for Type II entangled photon pairs.

Thank you! Things are clearing up. What do you mean by "know in advance which angles you are measuring"? Like, you need to know not only the difference in the pairs of angles, but the actual values of the angles in the pair? How would the values of the angles have an impact on the experiment separate from the impact of the simple difference between the angles?

Sorry, I don't yet have any formal education in QM, so this is all relatively new (and very strange) to me.

 

[Nugatory]

Thank you! Things are clearing up. What do you mean by "know in advance which angles you are measuring"? Like, you need to know not only the difference in the pairs of angles, but the actual values of the angles in the pair? How would the values of the angles have an impact on the experiment separate from the impact of the simple difference between the angles?

The difference between the angles is all that you need to see the problem.

Try this link: http://www.felderbooks.com/papers/bell.html, see if that help further clear things up.

 

[LikesIntuition]

I think my lingering question is this: why would something different happen depending on when two particles "decide" what property to exhibit? For example, why would two photons deciding how to be polarized when they reach two lenses be different from them deciding how to be polarized before they reach the lenses?

I feel like I'm overlooking something important. It seems to me like the same "options" would be available to the photons no matter when they collapse into a defined state. With QM, do they get to always pick a state where one photon definitely makes it through a polarized lens, or something like that?

Thanks for the help, everyone. Nugatory, I am reading the page you posted now.

 

[jk22]

When studying the covariance What we can write is obviously : $$cos(a-b)=\sum f_i(a)f_i(b)$$

With f1=cos, f2=sin it is a simple trigonometric identity.

But this is not equivalent to writing $$\cos(a-b)=\int A(a,v)B(b,v)\rho(v)$$

Since $$\int A(a,v)\rho(v)\leq 1$$ whereas $$\sum f_i(a)=\sqrt{2}$$

 

[LikesIntuition]

Nugatory, the page you posted helped A LOT. Thanks!

So, I understand now how the lack of objective reality would cause disagreements with Bell, but why would faster-than-light communication between particles do this? Using the example from that page (http://www.felderbooks.com/papers/bell.html), the electrons are released, the detectors are the same, one electron reaches its detector first and makes a decision, and this decision causes the other electron to make the same decision, right?

So, wouldn't a) the second electron carry on through its detector with that property or b) the second electron has changed by the time it gets to its detector, in which case what we care about wouldn't be spooky action at a distance, but the lack of a defined state for the electrons at any time other than when one or both of them are being measured?

I hope that makes sense. I will try to clear it up if need be.

 

[DrChinese]

I think my lingering question is this: why would something different happen depending on when two particles "decide" what property to exhibit? For example, why would two photons deciding how to be polarized when they reach two lenses be different from them deciding how to be polarized before they reach the lenses?

No one really knows why, and no one really knows how. That is why it is something of a mystery, which makes it all the more interesting.

 

[LikesIntuition]

So we do at least know that the data we actually collect disagrees with Bell's theorem, right? Then we make a conclusion that "spooky action at a distance" is at play. I can see why rejecting objective reality would make our worldview fit with the experimental results better, but why exactly do the results lead us to the conclusion of faster-than-light communication?

 

[DrChinese]

So we do at least know that the data we actually collect disagrees with Bell's theorem, right? Then we make a conclusion that "spooky action at a distance" is at play. I can see why rejecting objective reality would make our worldview fit with the experimental results better, but why exactly do the results lead us to the conclusion of faster-than-light communication?

Technically you only need to accept one or the other (reject realism or reject locality). Although sometimes it is difficult to draw a distinction.

 

[Nugatory]

So we do at least know that the data we actually collect disagrees with Bell's theorem, right?

To be precise, we know that the data that we collect disagrees with Bell's inequality. Bell's theorem shows that any theory in which the results of a measurement are determined by local and realistic properties (but see the note below) of the thing being measured must agree with Bell's inequality. Therefore, no such theory can be correct.

[note: Actually Bell's theorem does not speak of "local and realistic properties" - that phrase is rather too slippery to use in a mathematical proof. Instead Bell starts with a mathematical assumption about what variables could affect the results of a measurement

Then we make a conclusion that "spooky action at a distance" is at play. I can see why rejecting objective reality would make our worldview fit with the experimental results better, but why exactly do the results lead us to the conclusion of faster-than-light communication?

Again, that's not quite right. Bell's inequality applies to theories that are both local and realistic; a theory that is local but not realistic, or realistic but not local, or neither realistic nor local, can violate the inequality. Thus experimental results that show that the inequality is violated can be explained either by a faster-than-light influence traveling from the site of one measurement to other (that is, not local) or by abandoning the idea that a property not measured must still have a definite value (that is, not realistic).

 

[LikesIntuition]

Again, that's not quite right. Bell's inequality applies to theories that are both local and realistic; a theory that is local but not realistic, or realistic but not local, or neither realistic nor local, can violate the inequality. Thus experimental results that show that the inequality is violated can be explained either by a faster-than-light influence traveling from the site of one measurement to other (that is, not local) or by abandoning the idea that a property not measured must still have a definite value (that is, not realistic).

And I understand why the lack of realism would violate the inequality. But what's unclear to me at this point is exactly HOW faster-than-light influences would violate it as well. I'm still missing something...

 

[TrickyDicky]

And I understand why the lack of realism would violate the inequality. But what's unclear to me at this point is exactly HOW faster-than-light influences would violate it as well. I'm still missing something...

They violate the inequalities because they are non-local, and if you take relativiry seriously they are also unrealistic and impossible so I wouldn't worry about them.

 

[Nugatory]

And I understand why the lack of realism would violate the inequality. But what's unclear to me at this point is exactly HOW faster-than-light influences would violate it as well. I'm still missing something...

If there were a faster-than-light influence, then it could conceivably allow the measurement at one side to influence the measurement at the other in ways that the premise of Bell's proof of his theorem do not allow.

One of the premises going into that proof is that the probability distribution of results at one detector can be written as a function of the initial state of the pair and the state at the detector but not the state at the other detector; this is the locality assumption. However, if there were a faster than light influence traveling between the two detectors, then it would be possible for events at one detector to affect the probability distribution of the results at the other detector, invalidating that premise so that the proof will not apply to any theory that includes a superluminal influence.

One example of a non-local theory is naive wave-function collapse: I measure the state of one of the two entangled particles and the wave function of the whole shebang immediately collapses everywhere. It's not at all clear how that would happen, but no matter how it happens, "immediately collapses everywhere" can only happen if there is a faster-than-light influence. With tongue in cheek, you could imagine the measured particle sending a faster-than-light message to its counterpart saying "Dude, I've just been measured spin-up in the 120 degree direction; starting now you have to act like a spin-down-120 particle".

If by "exactly HOW" you're asking by what mechanism that faster-than-light message would operate, there's no good answer. The point of Bell's argument is that any local and realistic theory, no matter how it operates, will respect the inequality. It doesn't actually propose any theories, it merely tells us that whatever the winning theory is, it won't be local and realistic.

 

[LikesIntuition]

One example of a non-local theory is naive wave-function collapse: I measure the state of one of the two entangled particles and the wave function of the whole shebang immediately collapses everywhere. It's not at all clear how that would happen, but no matter how it happens, "immediately collapses everywhere" can only happen if there is a faster-than-light influence. With tongue in cheek, you could imagine the measured particle sending a faster-than-light message to its counterpart saying "Dude, I've just been measured spin-up in the 120 degree direction; starting now you have to act like a spin-down-120 particle".

Thanks! I think what still isn't clicking with me is this: wouldn't one of them have been spin up and the other been spin down even with locality? In a non-local theory, would the first particle to me measured have a different set of "choices" to make than in a local theory?

 

[DrChinese]

Thanks! I think what still isn't clicking with me is this: wouldn't one of them have been spin up and the other been spin down even with locality? In a non-local theory, would the first particle to me measured have a different set of "choices" to make than in a local theory?

No, that is the twist with Bell and a local theory. Here is the analogy on that. Imagine a single stream of photons with unknown polarization (such as from a light bulb). Not a stream of entangled photon pairs. Ask what the polarization (if measured) is at 0, 120 and 240 degrees (since you imagine there are values for these) for a handful.

0 / 120 / 240
+ + - (1 out of 3 match)
- + - (1 out of 3 match)
+ - + (1 out of 3 match)
+ - - (1 out of 3 match)
+ + + (3 out of 3 match)
etc

You cannot make it work out that the relationship between any pair of angles (any 2 of the above) is the same as the Quantum Mechanical prediction of 25%. You will always get an average of 33% or higher. That is because there is NO set of hidden variables that is consistent AND follows the QM relationship (cos^2 theta and know that for theta=120 degrees, the value is 25%).

So...

When you make your statement about one of the particles, you are assuming it has pre-existing values for any possible angle setting - and ditto for the other particle. But those values (for ONE particle stream) cannot match QM statistics. You probably won't ever see this until you try to write some data sets down as I have done above. There is little math in this process, as every time there will be either 1 match of 3 or 3 matches of 3 and NEVER 0. So you obviously could never average LESS than 33%.

Similarly there is no satisfactory dataset when you consider a data stream of entangled particles either.

 

[LikesIntuition]

When you make your statement about one of the particles, you are assuming it has pre-existing values for any possible angle setting - and ditto for the other particle. But those values (for ONE particle stream) cannot match QM statistics. You probably won't ever see this until you try to write some data sets down as I have done above. There is little math in this process, as every time there will be either 1 match of 3 or 3 matches of 3 and NEVER 0. So you obviously could never average LESS than 33%.

Very helpful, thanks! That makes a lot of sense. So somehow, faster than light transmission can account for an average lower than 33%?

 

[Nugatory]

Thanks! I think what still isn't clicking with me is this: wouldn't one of them have been spin up and the other been spin down even with locality? In a non-local theory, would the first particle to me measured have a different set of "choices" to make than in a local theory?

I found myself answering this question just a few days back in another thread - try this post https://www.physicsforums.com/showpost.php?p=4790812&postcount=57

 

[morrobay]

Nugatory, the page you posted helped A LOT. Thanks!

So, I understand now how the lack of objective reality would cause disagreements with Bell, but why would faster-than-light communication between particles do this? Using the example from that page (http://www.felderbooks.com/papers/bell.html), the electrons are released, the detectors are the same, one electron reaches its detector first and makes a decision, and this decision causes the other electron to make the same decision, right?

So, wouldn't a) the second electron carry on through its detector with that property or b) the second electron has changed by the time it gets to its detector, in which case what we care about wouldn't be spooky action at a distance, but the lack of a defined state for the electrons at any time other than when one or both of them are being measured?

I hope that makes sense. I will try to clear it up if need be.

I agree: Its more understandable to save locality and give up pre existing values (realism). That there is a
"lack of a defined state for the electrons at any time other than when one or both of them are being measured "
Physical values for the particles like spin and polarization are changing continuously until measurement.
This would be the basis for a local non realism model/explanation for Bell inequality violations.

 

[PeterDonis]

Its more understandable to save locality and give up pre existing values (realism).

Giving up "realism" doesn't give you locality, at least not as Bell defined it. No matter what you do, the correlations between spacelike separated measurements on entangled particles will violate the Bell inequalities, and that means there is *no* model that can both reproduce the observed correlations and also have the outcome at each measurement only be a function of the variables local to that measurement. "Realism" doesn't come into it, because Bell's theorem talks about what models can reproduce the observed correlations, and those are "real" even if the underlying properties of the observed particles aren't.

 

[LikesIntuition]

Does anyone think they could describe a situation in which giving up locality would violate Bell's inequality?

For example, using DrChinese's example above, how can we get less than 33% when we allow for non-locality? I can see how a lack of realism would do this, but I still don't see clearly how giving up locality would let us match QM.

 

[Nugatory]

Does anyone think they could describe a situation in which giving up locality would violate Bell's inequality?

Yes.
If you give up locality, than you can have a theory in which if one particle of the entangled pair is measured be up on a 120-degree axis its partner will become down-120; if the first particle is measured to be up-240 its partner will become down-240; and if the first is measured to be up-0 its partner will become down-0. In such a theory, Bell's inequality will be violated. This theory is also necessarily non-local; there's no way for the partner particle to "know" what state it should switch into without some non-local communication from the site of the first measurement to the site of the second. (And, at the risk of repeating myself, this is pretty much what the traditional collase interpretation says happens - we measure one particle and an instantaneous non-local influence collapses the wave function of the other).

You first thought will be to say that we don't have to give up locality to produce that effect; for all we know, the partner particle might have been created in the right state in the beginning. But that's not possible, because the states down-0, down-120, and down-240 are different states that can be experimentally distinguished - the partner particle cannot have been created in all of them at once, and if it were created in one of them then it would only be right for one of the three possible measurements of the first particle. Under those circumstances, where the second particle does not change its state in response to the measurement of the first, Bell's inequality cannot be violated.

You can convince yourself of this by constructing a sample data set; there are some examples in the link I pointed you at earlier.

 

[LikesIntuition]

You first thought will be to say that we don't have to give up locality to produce that effect; for all we know, the partner particle might have been created in the right state in the beginning. But that's not possible, because the states down-0, down-120, and down-240 are different states that can be experimentally distinguished - the partner particle cannot have been created in all of them at once, and if it were created in one of them then it would only be right for one of the three possible measurements of the first particle. Under those circumstances, where the second particle does not change its state in response to the measurement of the first, Bell's inequality cannot be violated.

That was absolutely my first thought. Do we have to give up realism and locality, or just one or the other? Aren't we stipulating that it's impossible for one of the particles not to "match" the other in both QM and using locality? It's never possible to have one particle be up-120 and the other be down-0, is it? And if that's the case, I still can't see why it matters when the particles decide what to do. But obviously, there's something about the non-locality that I am blatantly overlooking. Can anyone see what that is? Maybe I'm not fully understanding what non-locality is or something along those lines...?

Thanks again for the help. I'm learning a great deal, even if I'm still confused about this one thing.

 

[Nugatory]

It's never possible to have one particle be up-120 and the other be down-0, is it?

Of course it is possible. If I align the left-hand detector on the 0-degree axis and the right-hand detector on the 120-degree axis, there are exactly four possible outcomes:
1) left-hand particle measures up-0; right-hand particle measures up-120
2) left-hand particle measures up-0; right-hand particle measures down-120
3) left-hand particle measures down-0; right-hand particle measures up-120
4) left-hand particle measures down-0; right-hand particle measures down-120

I still can't see why it matters when the particles decide what to do. But obviously, there's something about the non-locality that I am blatantly overlooking.

Consider the ratio of case #1 above to case #2 above when we observe a large number of pairs. The quantum-mechanical prediction is that this ratio will be what we get when we pass a down-0 particle through a 120-degree detector, and this makes sense because our up-zero measurement of the left-hand particle tells us that the right-hand particle should be a down-zero.

We also have the four cases:
5) left-hand particle measures up-240; right-hand particle measures up-120
6) left-hand particle measures up-240; right-hand particle measures down-120
7) left-hand particle measures down-240; right-hand particle measures up-120
8) left-hand particle measures down-240; right-hand particle measures down-120
from when we have the left-hand detector set to the 240-degree axis. Again, quantum mechanics predicts and experiment confirms that the ratio of of #5 to #6 is what we get when we pass a down-240 particle through a 120-degree detector, and again this makes sense because our measurement of the left-hand particle tells us that the right should be a down-240.

But (and this is the key!) the two ratios #1 to #2 and #5 to #6 are different. The only way of explaining this result is if the right-hand particle, as it approaches its 120-degree detector, behaves differently when the left-hand detector is set to 0 degrees and when it is set to 240 degrees. Furthermore, we can wait until both particles are in flight before we decide whether to set the left-hand detector on the 0-degree or the 240-degree axis, so the behavior of the right-hand particle cannot have been determined when the pair was created - the right-hand particle has to change its behavior in flight as a result of the setting of the left-hand detector.

 

[Nugatory]

Do we have to give up realism and locality?

You have to give up at least one of the two.

 

[LikesIntuition]

But (and this is the key!) the two ratios #1 to #2 and #5 to #6 are different. The only way of explaining this result is if the right-hand particle, as it approaches its 120-degree detector, behaves differently when the left-hand detector is set to 0 degrees and when it is set to 240 degrees. Furthermore, we can wait until both particles are in flight before we decide whether to set the left-hand detector on the 0-degree or the 240-degree axis, so the behavior of the right-hand particle cannot have been determined when the pair was created - the right-hand particle has to change its behavior in flight as a result of the setting of the left-hand detector.

Alright. That is what I was missing. Thank you, that was extremely helpful!

 

[stevendaryl]

Of course it is possible. If I align the left-hand detector on the 0-degree axis and the right-hand detector on the 120-degree axis, there are exactly four possible outcomes:
1) left-hand particle measures up-0; right-hand particle measures up-120
2) left-hand particle measures up-0; right-hand particle measures down-120

...

5) left-hand particle measures up-240; right-hand particle measures up-120
6) left-hand particle measures up-240; right-hand particle measures down-120

...

But (and this is the key!) the two ratios #1 to #2 and #5 to #6 are different.

Hmm. What are the numbers for those cases? I thought that

Probability of both measurements resulting in spin-up = 1/2 sin²(Θ/2) = 1/2 sin²(60) = 3/8 (where Θ = angle between the two detector orientations).