# Are photons passing a polarization filter randomly?

entropy1
This may be a dumb question, but maybe someone can help me out:

Consider a pair of entangled photons A and B, fired at respectively Alice and Bob who both let it go through a polarisation filter at different angles. Now Alice establishes that half of the photons get through her filter, and half don't. So does Bob at his filter. Now, is this passing of photons at a single filter purely random?

Follow up question: however, since there is a correlation between the passing of photons by Alice and Bob, may we conclude that the event of a photon passing either filter is in fact not entirely random? (but perhaps seems random?)

## Answers and Replies

Mentor
Consider a pair of entangled photons A and B, fired at respectively Alice and Bob who both let it go through a polarisation filter at different angles. Now Alice establishes that half of the photons get through her filter, and half don't. So does Bob at his filter. Now, is this passing of photons at a single filter purely random?

Follow up question: however, since there is a correlation between the passing of photons by Alice and Bob, may we conclude that the event of a photon passing either filter is in fact not entirely random?

Not a dumb question, but you do have to use language more precise than "purely random". The two facts that you can take to the bank are:
- The probability that any given photon passes Bob's filter is 50%.
- The probability that any given photon passes Bob's filter given that Alice's matching photon passed her filter at a given angle is ##\cos^2(\alpha-\beta)##.

Using the language of probability theory, what do you mean by "purely random"? After you've answered that, you can decide whether it applies here.

entropy1
Gold Member
... (but perhaps seems random?)

Nugatory already answered, but I have this minor comment: how would you know if something appears random but is not? In my book, the only way to know is to find a root cause.

In the case of quantum particles, there is sometimes a hypothesized root cause. The problem with that is that none has ever been discovered. Barring such discovery, you are left with a tautology (you assume what you seek to prove). So it makes no sense to assert there is some root cause that explains "apparently random behavior" before you know what it is. If you want to look for it, fine, but short of that... "it appears random" is the same as "it is random".

Weddgyr
The behaviour of the photons at each individual polariser cannot be described by any probabalistic process or model, as the two-detector correlations of any such model fall within certain bounds that actually measured correlations break out of. I'm treating the polarizer/detector as a unified system here.

It would probably be more correct to say that the photons do not have any "behaviour" at all at each end. Detectors go off with certain correlations, but there is no element of physical reality which can be ascribed to their processes of going off.

Follow up question: however, since there is a correlation between the passing of photons by Alice and Bob, may we conclude that the event of a photon passing either filter is in fact not entirely random? (but perhaps seems random?)
Once the correlations have been measured, you may after the fact find the probability, or more correctly, the percentage of times a particular detector went off (50% for any setup). But you cannot ascribe this to the process of or probability of a photon passing an individual filter. I'm possibly being pedantic with language here, but here is an example which corressponds to two detectors with a relative inclination of pi/4 (45 deg).

Consider a standard 4 sided dice, on the faces( actually the peaks) of which are written the 4 possible left/right detector outcomes for the experiment: 00, 01, 10, 11. (0 - no detection, 1 - detection). You roll the dice to determine the outcome for the entire experiment. For example you roll and obtain the outcome "10". So the left detector goes off, and the right detector does not. In this case, ex-post facto, you interpret this as the left photon passed through its filter, and the right photon was blocked by its filter.

You perform this dice roll several times obtaining a series of results. Here's a dataset of 20 detector pair rolls I got the computer to produce.
11
10
11
00
00
11
01
10
11
10
00
00
11
01
00
01
00
10
10
10

ans =

0.55 0.40
The last line shows that in this particular set, the left detector went off (was "1") in 55% of cases, and the right went off in 40% of the 20 cases. As the number of cases increases, both these figures will tend to 50%. Now in this particular instance it is possible for you to produce a hidden variable model (both photons have a 50% of passing), but if I adjust the angle away from pi/4 this becomes more difficult(impossible?), and if I allow the polariser angles to vary randomly this becomes impossible.

The point I must stress here is that the dice rolls determine the outcome for the entire experiment. For a detector with left polariser at angle A and right polariser at angle B, you must construct a dice which gives the outcomes

:00 with probability $\frac{1}{2} \cos^2 (A-B)$
:10 with probability $\frac{1}{2} \sin^2 (A-B)$
:01 with probability $\frac{1}{2} \sin^2 (A-B)$
:11 with probability $\frac{1}{2} \cos^2 (A-B)$

Again I stress you are rolling to determine the outcome of the pair, not the outcome of a detection result at anyone polariser. If you perform enough dice rolls, overall each detector will go off (be "1") in around 50% of dice rolls. You cannot arrange the resulting table of probabilities as the product of a pair of single detector probabilities ($P_{11} \ne L_{1} R_{1}$), but you can express the single detector probabilities (percentages) as the sum of the the joint probabilities $L_{1} = P_{10}+P_{11}$.

My post will not make much sense, but I assure you that the result does not either. The probabilities L_1 and R_1 do not exist as independent probabilites. Only the joint probabilities are independently computable. I don't know what the formal term for this kind of relationship is, but whatever it's called it's the one borne out by the results of photon correllation experiments.

entropy1
The probabilities L_1 and R_1 do not exist as independent probabilites. Only the joint probabilities are independently computable.

If you have a photon-pair source P, then you can tell when there are photons fired, right? Then can you compare the number of photon-pair emissions with the number of detections of Alice on one hand (and independently Bob on the other)? That is to say, if you know you fired a photon pair, and measure behind Alice's filter, can you tell if the photon passed Alice's filter? (similarly for Bob's filter) So you can measure the probability (50%) of photons passing?

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Weddgyr
Timewise the detections themselves behave exactly as one would expect a local process would. You can arrange to have Alice's detections take place before or after Bobs, or indeed before or after ALL of Bobs detections. Before Bob had detected anything you will be able to measure the percentage of detections at A and it be about 50% for a large enough set. You can call this a probability of detection counts for many outcomes, but it will not be the probability of detection counts for a single outcome. There exists no such single detection probability or physical model.

If the question you are getting at is how a physical event can occur with a seemingly individual 50% probability but yet the joint probabilities (of a 11 detection) come out to be 1/2cos(A-B)^2, I don't have an answer and neither does anyone else except to say that is the way the joint probabilities will come out.

I've realized I've mixed up a lot of english words in my posts. By "event", I mean an actually detection occurring. Such events have "probabilities" (percentages) which can only be determined after experiments have been performed(I'm not a statistician). By "process" I mean some kind of mechanism or model for how a photon will pass through a polarized. Such processes have determinable "probabilities" which exist without actual events being recorded. The "50%" in EPRB type detector counts is the probability(percentage) of "events", not the probability of any kind of process. There is no kind of single detector process which can produce the joint detector probabilities recorded by experiment (events) or predicted by QM (processes).

Despite of course the fact that single detector events do occur and for all intents and purposes appear to occur locally, there is no single detector process local or non-local(*) which can generate them.

(*)To my understanding once special relativity is included, non-local models stop working as well.

entropy1
That is all pretty complicated. I'll try to put my follow-up question more simply:
• Is it possible to fire a pair of entangled photons on command?
• If it is, is it true that we can determine if photon A, one of the pair, passes Alice's filter or not? Similarly for B and Bob's filter.
• So now we have a single pair of measurements.
• We can repeat the measurements many times and find a 50% probability of passing a single, individual filter. Also, we can establish that there is a probability of cos^2(beta-alpha) that both photons of the pair pass.
I hope I am clear. Do I've got it right? Is it possible this way?

I guess I mean this: if there is no way of predicting if a single photon is going to pass the filter, we have to speak of a probability it does. But since there is a correlation between Alice's and Bob's measurements, there has to be a mechanism behind it, doesn't it?

Weddgyr
• Is it possible to fire a pair of entangled photons on command?
• If it is, is it true that we can determine if photon A, one of the pair, passes Alice's filter or not? Similarly for B and Bob's filter.
• So now we have a single pair of measurements.
• We can repeat the measurements many times and find a 50% probability of passing a single, individual filter. Also, we can establish that there is a probability of cos^2(beta-alpha) that both photons of the pair pass.

• Effectively yes. You can set things to go "one at a time" or mostly so.
• We can determine if they have passed a given filter, but we cannot determine whether they will pass beforehand
• The measurements are only after the fact of the events. They can't be determined individually before, but you can determine the probabilities(process) of the joint outcome beforehand.
• Yes and yes to both of these, in the sense that the measurements will confirm both probabilities (percentages of events) after the fact.

But since there is a correlation between Alice's and Bob's measurements, there has to be a mechanism behind it, doesn't it?
My instinct would be that there is, but the data would suggest there isn't. The point of Bell's theorem is to show that no mechanism or process which occurs at the detectors can explain the joint correlations. A lot of people assert that a non-local mechanism will work, but I have serious reservations about this due to relativity, but I'm not an expert on the matter.

Your instinct that there should be a mechanism of some kind at play at the polarization filters/detectors lies at the heart of the entire matter around Bell's theorem and entanglement in general. It sounds, no it is completely absurd to suggest that there is no physical probability or process taking place at each filter/detector, but yet no physical process can produce the joint correlations which are observed. There is no element of physical -- or mathematical -- reality which can explain why the detector beeps or not as a result of earlier emission.

This is totally daft and cannot be explained to anyone in any language short of a complete mathematical description. And it's obvious from the literature and continued investigation of these points that even experts won't necessarily accept the argument. But I digress and mope. My advice would be to code up a few simple models to see what happens. Do try coding up a non-local model as well. But at the end of the day you shouldn't expect to find a satisfactory explanation for what's going on with these entangled photon pairs.

entropy1