# Bell's inequality experiment

Hi. After spending an entire day watching videos and reading websites, I'm unable to find the details of a bell's inequality experiment with regards to photons. Could someone please describe such an experiment. What I've learned is that a photon is emitted, say blue. The blue photon is used to produce two red photons, which are entangled. The next part is confusing. I've seen experiments that pass each photon through various polarizers. While other experiments simply skip the polarizers and measure the horizontal and vertical polarizations. Are they trying to find both the horizontal and vertical polarizations? Why would the quantum results be any different than the results predicted by the hidden variable results?

Thanks for any help.

## Answers and Replies

Nugatory
Mentor
The next part is confusing. I've seen experiments that pass each photon through various polarizers. While other experiments simply skip the polarizers and measure the horizontal and vertical polarizations. Are they trying to find both the horizontal and vertical polarizations?
We'd have to see the specific descriptions that are confusing you to be sure what's going on. The idea is always to measure the polarization of one photon in one direction and the polarization of the other in one direction (possibly different, possibly the same) so that we end up with two measurements, one for each member of the pair. However, this is trickier than it sounds and usually involves multiple polarizers, slits, lenses, mirrors, and the like; chances are that the some of the descriptions you've read skip all these details and others include them.

Why would the quantum results be any different than the results predicted by the hidden variable results?
Do you mean "Why does the universe obey the rules of quantum mechanics?" If so, we don't know why it does, any more than we know why there's an ##r^2## in the denominator of coulomb's law instead of (for example) an ##r^3##. Science is a lot better at explaining how the universe works than why.

Or do you mean "Why do we say that the experimental results that agree with quantum mechanics are inconsistent with any local hidden variable theory?" For that, you have to look at the derivation of Bell's theorem. A good place to start is http://www.drchinese.com/Bells_Theorem.htm

• Symmetry777
DrChinese
Science Advisor
Gold Member
Hi. After spending an entire day watching videos and reading websites, I'm unable to find the details of a bell's inequality experiment with regards to photons. Could someone please describe such an experiment.

Here is a good reference with plenty of detail :-) :

http://arxiv.org/abs/quant-ph/0205171
"Entangled photons, nonlocality and Bell inequalities in the undergraduate laboratory"

• Ponderer

That's a well written easy to understand page. Although it didn't provide any detailed experiment. So it led me to a Wikipedia article on EPR, which help a lot.
http://en.m.wikipedia.org/wiki/EPR_paradox

Some other helpful pages :

http://en.m.wikipedia.org/wiki/Bell_test_experiments

http://en.m.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments

http://en.m.wikipedia.org/wiki/Quantum_mechanical_Bell_test_prediction

This photon experiment seems easiest to understand :

http://en.wikipedia.org/wiki/Bell_test_experiments#A_typical_CHSH_.28two-channel.29_experiment

I would like to know step by step examples of the above CHSH two channel experiment. It mentions for separate subexperiments with different settings: 0, 45°, 22.5° and 67.5°. Furthermore, it says, "Each photon encounters a two-channel polariser whose orientation can be set by the experimenter." Is this "orientation" referring to the four different angles (0, 45°, 22.5° and 67.5°) of polarization?

Let's analyze this experiment using Einstein's hidden variable interpretation. If a photon is at 0° polarization, the two-channel polariser orientation is set to 22.5°, and since it's a single photon I'm going to guess that ideally 100% of the photon will either be absorbed by the two-channel polariser, or it will travel through the polariser. When the two-channel polariser orientation is set to 45° and the photon is at 0°, then I'm guessing there's a higher probability of the photon being absorbed by the polariser than when the polariser is at 22°.

In one of those Wikipedia articles they state that if we know the photons z-axis polarization, then we can't know it's x-axis polarization. That doesn't make sense to me. My understanding of light polarization is that the photon is polarized at a specific angle when measured. Sure, you can get the two axises values from that angle, but if it's polarized at 0°, then how could there be any polarization in the 90° axis?

Thanks for clearing up any confusions.

DrChinese, thanks for the link. I'm still looking at it, but it might be too mathematical for me right now.

DrChinese
Science Advisor
Gold Member
If a photon is at 0° polarization, the two-channel polariser orientation is set to 22.5°, and since it's a single photon I'm going to guess that ideally 100% of the photon will either be absorbed by the two-channel polariser, or it will travel through the polariser. When the two-channel polariser orientation is set to 45° and the photon is at 0°, then I'm guessing there's a higher probability of the photon being absorbed by the polariser than when the polariser is at 22°.

In one of those Wikipedia articles they state that if we know the photons z-axis polarization, then we can't know it's x-axis polarization. That doesn't make sense to me. My understanding of light polarization is that the photon is polarized at a specific angle when measured. Sure, you can get the two axises values from that angle, but if it's polarized at 0°, then how could there be any polarization in the 90° axis?

Let me help on 2 different points:

1. An entangled photon has no pre-existing polarization. You are correct that it takes on a polarization when measured. If the polarizer is set at 22.5 degrees, then it will either be polarized now at 22.5 degrees or 112.5 degrees (which is 22.5 + 90 degrees).

2. A photon that is + polarized at 0 degrees is - polarized at 90 degrees. + and -, H and V, 1 and 0, etc are arbitrary labels that can be used interchangeably. A two-channel polarizer (PBS) lets + photons go straight through, while - photons are bent at an angle. Keep in mind that the angle settings have 0 degrees as an arbitrary direction usually.

0 degrees and 180 degrees are essentially the same. Ditto for 90 and 270. Any 2 angles 90 degrees apart are (for photons) called perpendicular, crossed, orthogonal or sometimes anti-parallel.

• Ponderer
Thanks DrChinese. With what you've shown me, I'd like to take another stab at this.

Two entangled photons are traveling in opposite directions, each toward a polarizer. Lets analyze one of the photons. It will go through the polarizer or it will reflect at an angle, for a total of two options. Two detectors are used to determine which path. After repeating this many times we get a percentage of how often it went to each of the two detectors. Next we rotate the polarizer and repeat the experiment. Again we rotate the polarizer and repeat the experiment. The end result shows that QM prediction is correct. How's that for an outline of the experiment?

DrChinese
Science Advisor
Gold Member
Thanks DrChinese. With what you've shown me, I'd like to take another stab at this.

Two entangled photons are traveling in opposite directions, each toward a polarizer. Lets analyze one of the photons. It will go through the polarizer or it will reflect at an angle, for a total of two options. Two detectors are used to determine which path. After repeating this many times we get a percentage of how often it went to each of the two detectors. Next we rotate the polarizer and repeat the experiment. Again we rotate the polarizer and repeat the experiment. The end result shows that QM prediction is correct. How's that for an outline of the experiment?

You're doing great! A couple of little add-ons:

1. The actual ratio at the 2 detectors by this polarizing beam splitter (let's call this PBS "Alice") is always 50-50 (very close anyway). When either detector "clicks" (a photon is detected), the time stamp is recorded too. The reason for 50-50: the outcome is random. :-)

2. The same thing happens at the other PBS, we will call "Bob". Again, 50-50.

3. Later, we compare the lists for Alice and Bob. We notice that there are clicks almost simultaneous for each (and a few we can make no sense of, let's ignore those for now).

4. Assuming we knew all along the angles the Alice and Bob polarizers were set at, we can compute the ratio: matches/(matches+mis-matches) at every pairing of angles. That will be a difference (Alice's angle - Bob's angle) usually of 0 degrees, 22.5, 45 and 67.5. Of course, you can also measure other ratios and you can also other angles. These are very common in experiments though.

Make sense so far?

• Ponderer, vanhees71 and bhobba
Oh I see. So the magic occurs in how often Alice and Bob get identical results at varying polarization angles. So at 0 (or 180) degrees Bob and Alice will always get opposite results. Example, if Alice's photon is up, then Bob's must be down. While at other angles they may or may not be opposite. After understanding the basics, videos like the following make sense.

Proof of Bell's theorem:

That's weird how the results are not as Einstein expected. But QM got it! In a video someone quoted a notable physicist as saying the instantaneous communication between the entangled particles occurs outside of spacetime, so there's no problem with relativity. Sounds like there could be some major breakthroughs in physics one day, hopefully soon. :)

• Symmetry777