Bell's theorem for dummies, how does it work?

Question

I've been reading up on theoretical physics for a few years now and I feel like I am starting to get an understanding of particle physics, at least as much as you can from Wikipedia pages. One thing I have tried to understand but fails to make sense to me is Bell's Theorem. I understand that the premise of the ERP paper was that the "wave form collapse" couldn't work because it would require the two particles which made up the convoluted wave form to communicate instantly, violating the information speed limit. Instead they suggested that it was due to hidden variables (ie the values are already set, whether they have been measured or not).

My question is can any one explain to me how Bell's experiment works and how it disproves this in terms that don't require an in-depth understanding of the math behind quantum mechanics?

My current understanding of the experiment is you have two people who are reading a quantum value of entangle quantum particles (for my understanding lets say the spin state of a positron-electron pair produced by a pair production event). For each particle pair the two readers measure the spin at a randomly chosen angle.

Here is where I need clarification: if I understand it correctly, local realism hypothesis states that when measured on the same axis the spin states should always be opposite (.5 + -.5 =0, ie conservation) when measure on opposite axis the spin states should always be the same ( .5 - .5 = 0 ) and when measured 90 degrees apart, the values are totally random. This part I get. I believe these results are predicted to be the same by both local realism and quantum mechanics. The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?

What I would like to have explained is the following:

1. What are the predictions made by quantum mechanics?

2. What are the predictions made by local realism?

3. How do they differ?

4. How is entanglement different from conservation?

5. Any corrections in regard to my explanation above?

Answer 1

Bell's theorem shows that standard QM is inconsistent with local realism. Local realism is a very general principle that was not originally thought to make any testable physical predictions. A major part of Bell's achievement was showing that Bell's inequality is implied by local realism, while standard QM predictions violate it. Experiments like Aspect's have since shown that Bell's inequalities are violated in reality, refuting local realism, in a way that is consistent with standard QM.

I think your issue is with the definition of local realism:

when measured on the same axis the spin states should always be opposite (.5 + -.5 = 0, ie conservation) when measured on the opposite axis the spin states should always be the same (.5 - .5 = 0) and when measured 90 degrees apart, the values are totally random.

This is just what standard QM predicts for entangled particles.

Local realism states that what happens at any point can only be directly affected by the state in its immediate neighbourhood, any long range effects must be mediated by particles or field disturbances travelling at (sub)luminal velocities, and that all behaviour is deterministic.

If entangled particles are far enough apart that one can perform measurements on both of them in a way that ensures the measurement events are separated by a space-like interval then local realism would require the particles to carry enough hidden variables to predetermine the outcome of each possible measurement, since any effect from one measurements would not have time to propagate to the other measurement to enforce the correlated observations.

Local realism and Bell's inequalities are not violated when only measurements separated by integer multiples of 90 degrees like in your description are considered. The discrepancy between QM and local realism only appears when oblique angles are considered, reaching a maximum when the angle between the measurements is 45 degrees (plus some multiple of 90 degrees), when the correlation between the measurements becomes $\sqrt{2}$ greater than allowed by Bell's inequality and therefore by local realism.

Spin conservation is really a separate issue. It just says that if the total spin of an isolated system was $x$ at some point in the past then it will always be $x$ and vice versa. Entanglemnt provides a way of satisfying conservation laws without assigning definite values of the conserved quantities to the individual components.

Bell's theorem is really about local realism and not really about QM. Experimental results could in principle violate Bell's inequality but not agree with QM predictions either. This would still rule out local realism and all theories satisfying it. The fact that QM does predict correlations higher than allowed by Bell's inequality and experimental results do agree with those predictions is kind of incidental.

Answer 2

To understand Bell's theorem it is not at all necessary to know anything about quantum mechanics. Essentially, it is sufficient if you believe that quantum theory predicts that it is violated even if the two measurements are space-like separated so that getting information what is measured at the other place is forbidden even by relativity.

http://ilja-schmelzer.de/realism/game.php gives a simple explanation how Bell's theorem works.

Bell proves first that, once both measure the same direction they get 100% correlation, but there cannot be information what has been measured on the other side, all measurement results should be predefined. Then he chooses three angles 0, 120 and 240 degrees. Assume now both measure different angles. Then we know two of the three values, all predefined, all + or -. Once out of three values + or - there is at least one pair equal, the probability of getting equal results should be at least 1/3.

Quantum theory predicts only 1/4 of getting equal results.

The straightforward solution, the one which is realized in the existing hidden variable theories like the de Broglie-Bohm interpretation, is that one of the hidden variables is a hidden preferred frame, and that the hidden variables can send information faster than light. But a hidden preferred frame, even if contradicted by nothing, is anathema in modern physics, and people prefer to reject realism, causality, logic and everything else, and fall into complete mysticism, only to avoid a preferred frame.

Answer 3

My understanding is that the measurement at 45° matches the measurements at 0° and 90° more than it should (assuming local hidden variables) given how often 0° and 90° match.

Think of two detectors that move between 0°, 45°, and 90°, so that you get the 90° measurement when one is at 0° and the other at 90° and the 45° measurement when one is at 45° and the other at either 90° or 0°. When measuring 45° and one of the other two angles, you get a match 85% of the time. So 90° matches 45° 85% of the time, and 0° matches 45° 85% of the time – how often must 90° and 0° match? At least 70% of the time – 0°, 45°, and 90° would all match 70% of the time, and for the other 30%, half the time 45° would match with 0° and half the time it would match with 90°. 45° would match either angle 85% of the time - 70% when all three angles match, plus 15% when 45° matches one but not the other.

But when 90° and 0° are measured, they only match 50% of the time. What’s the most that 45° can equally match the other two? 50% of the time all three match, then the other 50% of the time when 90° and 0° do not match, 45° can only match one or the other. If it matches one half the time and the other the other half of the time, the highest percentage you can get is 75%. 50% for when all three match, then 25% of the time matching 90° and not 0° and 25% of the time matching 0° and not 90°.

So to answer your questions:

1. What we actually see - 45° measurements matching 85% of the time, 90° measurements matching 50% of the time. This suggest that the angle of measurement of one particle has a correlation with the angle results of the measurement on the other particle.
2. Two separate things. If just looking at the results of 45° measurements, it says that 90° measurements must match at least 70% of the time (70% of the time when 0°, 45°, 90° all match plus 15% each for 0° and 90° when 45° matches one and not the other) . However, if looking at 0° and 90°, then it says that 45° can’t match the other two more than 75% of the time (50% when all three match plus 25% for each angle when 45° matches them and not the other).
3. Quantum predictions say that there can be a correlation between the angle of measurement of one particle and the result of the measurement on the other - even when there isn’t enough time between the final setting of the angle of measurement for one particle and the measuring of the other for light to travel between the two locations.
4. The correlation between the particles are connected to actions done to one of the particles.
5. I’d only argue with the wording of “completely random” for the 90° angle

I found this page useful for understanding the general concepts involved.

Answer 4

I will attempt to answer questions 1-3 as best I can. The others, I defer to the other excellent answers here provided.

Before I start: "The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?" -- correct.

2. What are the predictions made by local realism?

I think that this is really the crux of the problem, regardless of what the predictions made by quantum mechanics are. This is because Bell's Inequality does not state a prediction of QM -- it states a prediction of local realism (or of other sets of closely related philosophies, such as locality + counterfactual definiteness) -- and there is lots of evidence that the prediction of local realism made by Bells' Inequality does not hold. Thus, what QM predicts is only relevant if you are interested in one of its many interpretations to replace local realism. Of course, via these same experiments, the results tend to match the predictions of QM, so they also provide evidence for the QM equations, but I think that this not the main purpose of Bell's Inequality.

Bell's Inequality is a very abstract statement designed to cover any local realism theory. So, since intuition is what we are after, allow me to propose a particular local realism theory, which the experiments for Bell's Inequality will therefore provide equally good evidence against:

Hypothesis 1: The spin of a particle is governed by a hidden variable $\theta \in [-\pi,\pi)$. Denote $\theta_\phi$ to be the outcome of measuring a particle's spin when our measuring equipment is calibrated at an angle $\phi$ (so for all $\phi$, $\theta_\phi = 1$ or $\theta_\phi = -1$). In other words, even though we always measure spin to be up or down, there is some "hidden variable", $\theta$, that is a continuous-valued spin that is the "real variable" which is the "true" spin, we having only a poor window with which to view it, namely, $\theta_\phi$. For the sake of concreteness, we hypothesize the following behind-the-scenes mechanism for our measuring equipment:

$$\theta_\phi = \text{sgn}(\sin(\theta-\phi))$$

(this is a square-wave by the way... in a sense, you just round the angle you are measuring relative to the angle of your equipment. e.g. if $\phi=0$ and $\theta$ is negative, you get "down" and if the $\theta$ is positive you get "up")

Note that Hypothesis 1 gives us a mechanism for local realism since a particle has a definite spin (realism) given by $\theta$. Also, an explanation for correlations between measurements when we study pairs of particles at a particular "angle" can now be explained by a local property, $\theta$.

The next few paragraphs operate under the assumption of Hypothesis 1.

Now, per the usual example, let's generate sets of pairs of particles with opposite orientations. Let's just focus on a handful of pairs that we managed to generate, and let's pretend that we can peek under-the-covers to see: $\{(\theta_1, \theta_2)\} = \{(\pi/4,5\pi/4), (4\pi/3,\pi/3), (0.001, 0.001+\pi)\}$. We want to think about what happens when we measure these particles. Calibrate detector A at $\phi=0$ and detector B at $\phi=\pi$. If we split up the pairs, and send $[\pi/4,4\pi/3,0.001]$ to detector A, and $[5\pi/4,\pi/3,0.001 + \pi]$ to detector B, what do we expect to get? Plugging stuff into the above formula, we expect to get $[1, -1, 1]$ at detector A and $[1, -1, 1]$ at detector B. Play with the calibrations at detector A and B and re-plug stuff into the above equation. Note that no matter what you set them at, as long as they are opposite ($\phi_\text{A} - \phi_\text{B} = \pi$), then we get identical results at detectors A and B (although perhaps not the exact sequence $[1, -1, 1]$, depending on the calibration).

Now, notice that if we change the calibration of only A by a very small angle $\phi_\text{A} = 0.002$, then the value of the third particle in our list $a_3$ at detector A will flip. The corresponding measurement at B should not, because we have not altered its calibration, and $\theta$ for $b_3$ remains the same. In other words, if we don't change the calibration of B $\phi_B$, and we don't change any of the $\theta$ of our particles, then it is of no consequence what is happening over at A, whether researchers are measuring particles, or whether they have all gone to have a beer, what we measure at B is totally unaffected and has only to do with the calibration and particles at B. This statement is a necessary condition for local realism to hold. If, somehow, the corresponding measurement at B changes depending on whether an observation occurred at A, then either it communicated with its pair particle over at A (to change its $\theta$), or some other implicit assumption of Hypothesis 1 has fallen through. So, one prediction of our Hypothesis 1 is that the measurement for $b_3$ remains the same whether or not we make a measurement at $a_3$. If we can show that this does not hold, then Hypothesis 1 does not hold.

The exact situation when we expect Hypothesis 1 to fail, due to the predictions of QM, are kind of strange. If we measure particle 1 at B, then its partner particle 2 at A, and the re-measure particle 1 at B, QM does not expect the measurement at B to change. We only expect the measurement at B to be "altered" if we look at A first. This makes observing the supposed "alteration" difficult!

However, Bell proposed the following experiment by which we can test Hypothesis 1 (and a whole class of related hypotheses). If we generate a boatload of particle pairs according to a common general scheme, and then re-calibrate A and B to various convenient values, we can predict the probability of various observations at B both with and without having "looked" at the particles at A.

Here's the setup: Generate a very large quantity of particle pairs with the first particle having uniformly-distributed $\theta_1$, and the second having an opposite orientation $\theta_2 = \theta_1 + \pi$. We can test uniformity by simply calibrating our measuring apparatus at random locations and making sure we get an approximately equal number of "ups" and "downs". The only way for this to happen is if the $\theta_1$ are uniform. We can test that the two particles are always opposite by checking that, when A and B are calibrated at $\phi_\text{A} - \phi_\text{B} = \pi$ apart from one another, we always measure identical readings for each particle in a pair. Set $\phi_\text{A} = 0, \phi_\text{B} = \pi$. Alter $\phi_\text{A}$ (and only $\phi_\text{A}$) by a little bit. Generate another bunch of particle pairs. Now, some quantity of the particle pairs will not produce identical measurements (like our $a_3, b_3$ above). Write down this quantity $x$. Just to double-check, reset $\phi_A$, and alter $\phi_B$ by that same angle. Generate a bunch more particle pairs using the same mechanism. You should see that the number of unequal measurements is approximately $x$, because the situations are symmetric (but not exactly equal, because our $\theta$ are random). Just to quadruple-check, do this a whole bunch of times to convince yourself that the number of unequal measurements is pretty much always around $x$.

Here's the expectation: Now, change $\phi_\text{A}$ and $\phi_\text{B}$ by that small angle. In order for the problem to appear, we need to consider what might have been had we not altered $\phi_\text{A}$ or $\phi_\text{B}$ or both. If we hadn't altered either, because the measurements are all governed, under-the-covers, by $\theta$, we would have measured identical values for all pairs. If we had only altered one or the other, we would have measured different values for $x$ pairs. If we alter both, even if none of the pairs that "change" overlap, we measure different values on $2x$ pairs. Namely, all of the pairs whose measurements "changed" at B plus all of the pairs whose measurements "changed" at A. For the remaining pairs, since their measurement didn't change at A and didn't change at B, they still give identical measurements. If there is any overlap in which pairs flipped measurements at A and B, then the number of pairs giving different measurements will be strictly less than $2x$. To reiterate, this expectation only holds if the measurements at A and B do not effect each other. It also only holds if it is meaningful to speak of "what might have been". If the simple act of observing the spin of particle 1 at A changes the value of $\theta_2$ of its partner at B, then the situation where we make measurements at A and B need not have this particular relationship to the situation where we make only a measurement at A. For example, the act of measurement at A could change all of the $\theta$s at B to be totally random. Or it could change the $\theta$s at B to be the number predicted by QM. The only important thing here is that if A and B "talk", then the number of "different" measurements might be $>2x$.

At this point, it is worth noting that the exact mechanism we proposed above is irrelevant to the argument as a whole. You can replace all of the talk of "$\theta$" and the mechanism we proposed by which it is measured by talk of some "arbitrary locally-real variable encoding the spin info" and the inequality still holds.

1. What are the predictions made by quantum mechanics? & 3. How do they differ?

Basically, QM predicts that for certain calibrations of the equipment at A and B, we will reliably observe $>2x$ pairs that now give different measurements when we alter both $\phi_\text{A}$ and $\phi_\text{B}$. How much different depends on complex maths that are above my pay grade. If anyone in the community has a link to a location with an explanation of this math, please comment and I will edit it in.

However, as I said above, it's largely irrelevant to Bell's result what those predictions are. Simply performing the experiment and noting that the number of pairs with different measurements is $>2x$ is enough to reject local realism, even without anything to replace it with.

Somewhat orthogonal to the predictions made by QM are the available interpretations of this result now that local realism has been tossed. This answer to a related question provides a discussion of how these interpretations tie into the results from Bell's inequality.

Answer 5

(1) Quantum mechanics predicts 25% or more will correlate. (2) Bell says hidden variables should correlate 33% or more of the time. (3) The difference between the two is Bell's inequality. (4) They are different things unless I misunderstand your question. Two objects are entangled if you can measure or observe one of them and instantly know something about the other one. Conservation could mean some physical quantity in an isolated system is constant. (5) Your description of local realism seems complicated. My understanding is that a particle cannot get its instructions from a distant source that would take faster than light communication. Instead the particle most likely carried the instruction from its beginning.

Answer 6

I will use polarised photons as they are simplest to understand. Also to avoid confusion, we will assume both entangled photons have the same polarisation.

1. What are the predictions made by quantum mechanics?
2. What are the predictions made by local realism?

Below is a plot of the QM prediction (blue) and the local realistic model (red) correlations (y axis) versus the difference in angle between the two polarising analysers (x axis):

The quantum prediction for the correlation is equivalent to a single particle having passed through both the polarisers of the observers in the positions they are at, at the exact time the particles are received and is numerically equivalent to the equation for Malus Law ($\cos^2(\Delta \theta)$) which predicts the amount of light that is transmitted trough two successive polarisers with an angle of $\Delta \theta$ between them. Since Malus Law can be represented as ($\cos^2(|A-B|)$) where A in the the Angle of Alice's polariser and B is the angle of Bob's polariser, the local realistic model has the impossible task of predicting the outcome of that equation without knowing both A and B.

The implication is that each particle somehow 'knows' the position of the other polariser at the time the other particle passed though it, even though they may be separated in such a way that there is not enough time for any signal travelling at the speed of light to communicate that position. This implies non local interaction.

Bell's Theorem basically shows that any theory that relies on pre-coded information (however well hidden) and/or information passing that is limited to the speed of light, can not match the predictions of Quantum Mechanics and this is confirmed by the results of real experiments.