# Trouble with QM's theory on Bell's discovery

1. Jan 6, 2009

### Gothican

I've been reading Brian Greene's book 'The Fabric of the Cosmos', and had a little trouble in the part where he explains bell's discovery regarding the EPR paradox.

Bell's discovery in short was that there is a way to check if a particle has a definite spin about more than one axis or not; He prepared two detectors which measured the direction of the spin of two identical particles on three different axis', each 120 deg. apart (CW or CCW). Therefore, there are 9 different combinations of axis' the two detectors measure: (1,1),(1,2),(1,3) ... (3,3) while each number stands for the 3 different axis.
Now, according to EPR, the particles have definite spins about every axis, and the both of them are "programmed" before they leave the source. -> the universe IS local.
Therefore, the two detectors should find compatibility with the two particle's spins when they check the two particles about the same axis ((1,1),(2,2),(3,3)), and two more times because there are three axis' and only two different results - CW and CCW ((1,2),(2,1)). - 5 times out of 9.
Hope you guys got the picture.

Now my question is, what is the quantum expectation in this experiment?
I know that it's 50 percent of the time (as opposed to more than 50 per. according to EPR), but why?
What happens to the two particles if you check one's spin on one axis and then the second on another? If QM tells us that one of the particles makes the other immediately fixate upon one axis, and jump out of it's state of uncertainty, you shouldn't be able to measure the second's spin at all! Then what does the second detector find?

Gothican

2. Jan 6, 2009

### DrChinese

Welcome to PhysicsForums!

Of course you can always measure the spin of the second particle (usually called Bob). The question is, what does that have to do with the first (called Alice).

Mathematically, it does not matter whether Alice is measured before Bob or vice versa. The effect is the same. According to the QM state description, the photons exhibit "perfect" correlation when their spins are measured identically. That matches the EPR view too.

But QM predicts that when Alice and Bob are measured at different settings, their correlation is related to the cos^2(theta) where theta is the angle between. For 120 degrees, the cos^2(120)=.25 (1/4). So there should be 25% correlation. That is based on the idea that any particle polarized at the angle of Alice will have a 25% chance of also being polarized at the angle of Bob. Malus's Law, circa 1807.

That is at odds with standard probabilty theory for +/-120 degrees, which should be 1/3. In other words, considering that the three positions are 120 degrees apart (total 360, a full circle): the three cannot all have a 25% chance of being like their neighbor. There wouldn't be internal consistency. Try randomly flipping 3 coins and then randomly looking at just 2 of them. You will quickly discover they are alike (correlated) at least 1/3 of the time.

If either of these values (1/4 prediction for QM or 1/3 for probability theory) are not clear, let me know and I will explain further. However, suffice it to say that when the experimental test is run, the actual answer is 25% and not 1/3. I hope this helps.

3. Jan 7, 2009

### Gothican

Yeah, and in total 50 percent of the time:
1. When the two detectors measure the same axis -3/9 = 1/3
2. As you said -
when the two detectors measure different axis': 6/9 of the time. Now 6/9 x 25% = 1/6
3. 1/3 + 1/6 = 50%.

And thanks for the mathematical explanation, but what I was expecting was a more theoretical one.
I understood that QM is different than probability theory, but why?
What happens to Alice and Bob when the detectors measure them on different axis'?
And why is it at all connected to the angle between them?
I bet it's a pretty complicated answer, so if you guys aren't up to writing it all down here, it'll be great if someone could post a link to another page which explains it more fully.

thanks

4. Jan 7, 2009

### DrChinese

1. Probability theory assumes that the measurements on Alice and Bob are independent events mathematically. But any such version that shows "perfect" correlations (to match experiment) cannot help but run afoul of Bell's Theorem. This is because the classical (and common sense) notion of "realism" leads to the assumption that there are well defined values for particle spin at settings OTHER THAN at Alice and Bob. After all, if Alice and Bob are always "perfectly" correlated when measured at the same angle, AND they are independent observations, then you would have to be able to imagine well defined values at ANY setting - even if that setting is never actually observed. It is this requirement - called realism - which formed the basis of Bell's Theorem.

2. A measurement of Alice puts Bob into the identical (although random) state as Alice (and vice versa). Once Bob is in that state, a subsequent measurement of Bob will follow Malus's Law: cos^2(theta). Malus applies to any photon in a known polarization state. Then after Bob was measured at that different angle, Bob is in a different state than Alice.

3. So you can see that Alice (after her measurement) now represents the state of Bob before Bob was measured, and Bob now represents the state of Bob after his measurement. And that is how the entangled photons get their cos^2(theta) relationship.

2 and 3 above are, of course, the quantum mechanical explanation. It matches experiment. The thing is, the QM version of the story does NOT include the idea that the photon had a well-defined polarization at any possible angle independent of actually observing it (realism), while the "common-sense" realistic version does include that assumption.

Bell showed that no realistic theory (per 1 above) could ever match experiment - unless there were faster than light influences at work. Does this help?

5. Jan 7, 2009

### DrChinese

Here are a couple of my web pages on Bell's Theorem which run through the math:

Easy Math via Mermin's Approach: This uses the 120 degree example, which I think is easiest to follow. I shift it to exclude the cases of perfect correlations (where Alice and Bob are observed at the same setting) because that is actually a separate experimental requirement which alternative theories must match as well.

Bell's Theorem and Negative Probabilties: This shows that some subensembles have negative probabilities if you assume classical probability theory.

Both of the above are mainstream analysis following standard treatments.

Last edited: Jan 7, 2009
6. Jan 7, 2009

### DrChinese

LOL, the reference has nothing to do with the what the OP is asking, and it is intellectual dishonesty to state otherwise. And of course I checked the reference first.

By the way, have you checked out the reference I provided? You can take that, the Jaynes reference, and plenty more and START A NEW THREAD to discuss what you are interested in. I promise you will get the attention you want there. I am not going to debate your ideas in this thread, and that is the last comment I will make on it.

To the Original Poster: ignore mn4j's comments and you will be happier for it. There are also people out there that don't believe dinosaurs ever walked the earth, and there are people who think the Easter Bunny is real. 'nuff said.

7. Jan 7, 2009

### Gothican

It's funny to see other people arguing over a topic I never intended to raise in the first place. lol...

Well, anyway, I think I got a satisfactory answer from DrChinese;
1. Unlike regular probability, QM states that one observation changes the state of the second particle (as in DrChinese's explanation in post #4 - thanks!).
2. The mathematical equation to determine whether the second particle will have the same spin as the first (cos^2(theta)), is just a formula that "gets along" with reality. There is no realistic theory that can explain the formula.

Which means, according to Heisenberg's uncertainty principle, that before we measure Bob, all his spin is concentrated upon one axis - Alice's one.
Then, when we measure Bob on a different axis, all of his spin concentrates upon the one we measured, and he doesn't have any other spin on the other axes (anti-realism) because we can't measure them (uncertainty). Even though he had another spin "before".

If one of these understandings aren't true, it'll be nice if someone could fix them.

Thanks a lot from you all,
Gothican

8. Jan 7, 2009

### ZapperZ

Staff Emeritus
Please note that unless the post directly addresses the topic brought up by the OP (Gothican), it will be deleted as being an attempt at thread hijack. Please restrict the discussion to the confines of what is directly relevant to the topic.

Zz.

9. Jan 7, 2009

### JesseM

Do mods have the power to split posts off into a new thread? I could just copy and paste the discussion about the paper mn4j references into a long post in a new thread but it would probably look kind of ugly...

10. Jan 7, 2009

### ZapperZ

Staff Emeritus
Let me try....

Zz.

11. Jan 7, 2009

### DrChinese

Yes, you have it.

The interesting thing is that it doesn't matter for either of these 2 things:

1. The HUP limits our knowledge even for systems of 2 entangled particles to the same knowledge that we could have with one unentangled particle. In other words, the idea of the 1935 EPR paper - while ingenious as a way to get around the HUP - was ultimately wrong. You cannot use entanglement to beat the HUP.

2. The results are the same - mathematically at least - regardless of whether you think of it as Alice taking on Bob's orientation or Bob taking on Alice's orientation. Either way, the statistics will be the same.

Please follow up with any questions as needed, and I am sure someone will follow up as best possible.

-DrC