Bell's Inequality

Hello today I want to see what is called

bell

inequality

John Stuart Bell 1964 was responsible for this it is an answer to the Einstein-Podolsky-Rosen paradox about which I produced a video recently now just to review what that paradox said what Einstein said and Podolsky and Rosen argued that if you have some kind of system like nuclear energy or reaction, whatever it is, that is responsible for the production of pairs, you could produce, say, an electron and a positron, but what you discover experimentally is that if you measure the spin of the electron say against the z axis, the vertical axis, and you will find that the spin is up, you might not, you might find that it is down, but if you find that it is up, then what you will always find is that the spin of the positrons will be downwards, in other words, those two spins. will always be exactly opposite similarly, if you measure the spin along the x-axis and find that it is in that direction then if you were to measure the spin along the x-axis of the positron it will always be in the opposite direction now the principle of Heisenberg uncertainty says that you cannot simultaneously know the spin in the Z direction and the spin in the and what can be said is that when this pair of electrons and positrons is emitted, Alice will measure the spin in the Z direction and find, let's say, that it is vertical, then she yells at Bob.

I have found a turn in the Z direction that is up if you measured your turn, we know that. which you would find is down, but don't do that, instead measure your spin in the x direction, so Bob is supposed to measure a spin spent in the x direction and find that it is to the right, then he would know that Its turn in the Z direction was down and its turn in the X direction was to the right. He would know both values ​​at the same time, but Heisenberg says he can't, so in practice what would happen is that if Alice measured the spin in the Z direction of this electron, Bob.

I can't measure the spin in the X direction of the positron, but if Alice doesn't make that measurement, then Bob can make the measurement in the No? We made a measurement here and said that there were two possible explanations: one is that the electron and the positron are in constant and immediate communication, no matter how far apart they are, no matter how far apart they are, they are communicating with each other in a network. intertwined. Instantly tell what is happening but of course there is a problem with that which would suggest that information is transmitted instantaneously and is faster than the speed of light and special relativity says that is not possible, that is what called a violation of locality, the second explanation is that when these two particles were first produced when they were born, if you will, each one was embedded with some kind of DNA or hidden variables that would govern forever precisely what they would do in the entirety of their life and those hidden variables would ensure that they did.

They will always remain entangled because that DNA would always govern what they did now, since it was thought that locality could not be violated, in other words, you could not have constant communication of an immediate nature because that would violate Einstein Podolsky special relativity. and Rosen thought that there must be hidden variables embedded in these particles that would dictate what they did and it was Bells'

inequality

that shows that no such hidden variables can exist. Now, to understand how Bells inequality works, let's first look at a very simple example. We are considering children playing on a playground some of them are wearing hats some are wearing scarves and some are wearing gloves it is a cold day all of them are wearing at least one of those items of clothing some are wearing - some are wearing all three and we can represent those children by circles overlapping, they are sometimes called sets this is set A this is all the children with hats this is set B all the children with scarves and this is set C all the children wear gloves now where a and B overlap, that is this area here, that means kids wear hats and scarves. where a and C overlap, that's this area here, kids wear hats and gloves where B and C overlap, that's this area here, that means kids wear scarves. and gloves and right in the middle where B and C overlap, that means those kids wear hats, gloves and scarves and what

bell

inequality says is that all the people who wear hats but not scarves plus all the people who wear scarves but no gloves is always greater than or equal to all the people who wear hats but no gloves and I will demonstrate this by simply numbering each of the one, two, three, four, five, six, seven, so if you say it is not B , well, it's not B it's region one plus the region for two and five are part of B so one plus four is a not b what is B not C well B not C is going to be three plus two because five and six are part of C so this is plus three plus two and that says that bill has to be greater than or equal to zero C well, zero C is 1 plus 2 because 4 and 5 are part of C, so now we have 1 plus 4 plus 3 plus 2 must be greater than 1 plus 2 and that is obviously true because we already have 1 plus 2 on this side, let's put the 1 plus 2 there plus 4 plus 3 is greater than or equal to 1 plus 2 well, it will be equal to if not There are children in the categories. 4 or 3 but if there are children in categories 4 or 3 what is 4 4 is that they wear hats and gloves but not scarves 3 is that they wear scarves but not hats or gloves if there is someone in that then this equality or another inequality will be true , by the way, at this point.

I recognize Professor Leonard Susskind, who does something very similar to this in his videos on quantum entanglement. Now how does this help? Well, we must think, for example, about light, let's take light that can not be polarized, that is. To say that electric and magnetic fields can oscillate in any direction, you can pass that light through what is called a polarizer and a polarizer will polarize the light in only one direction. Suppose the polarizer will polarize the light in the upward, downward direction. direction so that the light passes through the polarizer and comes out the other side and is polarized in the upward direction.

Now suppose we take unpolarized bits of light and pass them through a polarizer in an upward direction. It comes out the other side polarized. Now we pass it. a polarizer that is exactly 90 degrees to the vertical polarizer, which will happen, the answer is that nothing comes out the other side, but if we take the same unpolarized light, we pass it through a polarizer in the vertical direction and then We pass it through a polarizer in the vertical direction. At a 45 degree angle, some light wouldn't come out as much, but some would, and if we then passed it through the polarizer that was horizontally polarized again, the sunlight wouldn't come out as much, but some would, so part of The light will stop as it goes.

In this polarizer, more light is stopped as you pass through that polarizer and more light is stopped again as you pass through that polarizer, but the point is that after you have passed through a vertical polarizer and a horizontal polarizer, you still have some left of light because you had this polarizer that was at an angle in the middle, whereas if you had done it this way nothing would come out right, it's an experiment, you can do it and you can demonstrate it, but now let's look at it from the point of view of an individual photon . a photon that could be fed polarized in any direction in practice will of course only be polarized in one direction, we just don't know which one and when it reaches the polarizer it will pass or not. it could be part of the photons that stop and it could be part of the photons that pass similarly, here a photon appears if it passes through this polarizer, okay, it is polarized in the vertical direction, but it definitely won't pass through. that polarizer, but here is the strange thing, here comes our same photon, it passes through this polarizer and it is polarized in the vertical direction, then it either passes through or it doesn't, there is a probability that it does or doesn't, it passes through this polarizer. , it is then polarized in the direction of the 45 degree angle and then when it reaches this polarizer, it might pass through or not, and if it does, it will polarize in the horizontal direction and whether or not it passes through those polarizers is a phenomenon of quantum mechanics.

I can't predict which of those photons will pass through, some will and some won't, so now let's try an experiment and this is again an entanglement experiment. We are going to have some process that produces pair production. In this case, I am going to produce two photons that shoot in opposite directions to Alice and Bob's laboratory, who will have polarizers and will have only three types of polarizer, one is the polarizer that is vertical, another is a 120 degree polarizer and one is which is a 240 degree polarizer, now it is important to note that you cannot predict what the polarization of these two photons will be, it could be in any direction, but if it is pair production, they will be the same, so if If two photons are produced, they will have the same polarization, if you then produce another two photons, they will have the same polarization but not the same as the previous two, if you understand what I mean, what we are going to do in this experiment is that we are going to produce pairs constantly. of photons what we know is that each pair will have the same polarization but the next pair will have a different polarization.

What we are going to do each time we produce a pair is that Alice and Bob will randomly select each one. They have these three polarizers and they will select one of them at random and see if that photon passes through the polarizer or not. According to Einstein Podolsky and Rosen's paradox, these photons are supposed to contain hidden variables, which is information that will tell them. how to behave in whatever circumstances they face, so presumably they must contain information that tells them whether or not they will certainly pass through these polarizers because Einstein didn't like the idea that it would be probabilistic, he wanted there to be certainty in all This he once said .

I like to think that the Moon is there even when I'm not looking at it. There is nothing probabilistic about this. He says that we must know that someone must know that the photon must know whether it will go through these polarizations or not. that is a hidden variable, so one of the hidden variables will be: you will go through polarizer one, you will go through polarizer two, and you will go through polarizer 3, and yes there are hidden variables, although we don't know what they are because we don't have them. We haven't discovered them yet, so you absolutely have to determine how a photon will behave if it encounters one of those polarizers and we found that there are eight combinations of this information, we can say that the first combination is if we take, for example, the three polarizers one two . and three we can say that the hidden information can say that it can go through all three.

The second option is that the hidden information says that it can pass through one and two but not through three. The third option is that it can go through one and three but not the fourth. The possibility is that you can get through one, but not two or three. The fifth possibility is that you can't get through one, but you can get through two and three. The sixth possibility is that you can't get through one or three, but you can. Moving on to the seventh possibility is that you can't get through one or two, but you can get through three and there are chances that you won't be able to get through any of them and those are the only bits of hidden information that could be contained in relation. about whether or not those photons will pass through any of those three polarizers if you encounter them and don't forget what we said, but every time you do a pair production you're going to get different hidden information, so that's how it will be.

It won't be the same every time. The production of the first pair could have this. The second player's production could have this. The third one might have this. You don't know, but what you do know is that each of those photons will have the same variations because they will both contain the same hidden variables. Now let's look at the experiment that Alice and Bob do. Each of them will randomly take one of their polarizers and see if their photon will pass through it, so the combination could be a and B or B. and C or a and of C Now, what will happen to each of these eight variations?

Remember that we don't know what hidden information photon we have. All we can do is do the experiment right. What we want to ask ourselves is when. Alice and Bob get the same information, the same result with their photon and when does it differ well if one has an A and the other is polar? So in each case, the photon will pass because its hidden information says it will. pass, sorry, this should not be an A and a B should be a polarizer 102 to 2 and 3 or 1 and 3, we know that in this, if the photons have this hidden information, they will go through 1 and 3 sorry, 1 and 2, then they will be equal, they will go through 2 and 3, so they will be equal and they will go through 1 and 3, so they will be equal.

This is not very well written, is it? 1 and 2, this is 2 and 3, this is 1 and 3, are the combinations of the polarizers that Alice and Bob have used for the particular couple production that they do and it interrupts me at this pointbecause after watching the video I think I already have. I haven't explained it as clearly as I'd like, so I'm going to make it a sort of explanation within an explanation and let's go a little slower. What we are saying is that we are going to have a production of pairs of photons. Some process will generate two photons and those photons will move in opposite directions toward the laboratory. from Alice and Bob who can make measurements on them and what we have determined experimentally is that although we do not know what the characteristics of the two photons will be in terms of their polarization, they will both have the same polarization if we do the experiment.

Again we will generate another two photons, those two photons will have the same polarization with each other, but not necessarily the same polarization as the previous pair. Now what I have said is that each of them has three polarizers, one polarizer that will let a photon pass through. in the vertical direction in the 120 degree direction and in the direct direction of 240 degrees and Alice and Bob will randomly select from their polarizers, place them in front of the photon and see if the photon passes through or not. It doesn't go through those are the only two options that you can see what goes or doesn't go through, of course there may be times when you both take the same polarizer.

Now there is no experiment in this regard. We know that if they take the same polarizer, the photons will pass through both or they will not pass through either because they are both polarized in exactly the same way, so if they pass through a virgin it will pass through a vertical polarizer, this will also do so at any time than Alice and Bob If we choose the same orientation of the polarizer, we can ignore the experimental result because that will be the case. We already know that is the case. Only when Alice and Bob choose different polarizers do we get information and what we want to record for each experiment every time.

We do a pair production, we're going to do this experiment where Alice and Bob randomly select between their three polarizers and see if the photon passes through the question we want to ask: do you get the same result or a different result right now? The result means that the photon passes through both polarizers or the photon does not pass through either polarizer. The same result if both pass or neither pass a different result means that a photon will pass through one polarizer but not the other. I hope that makes what follows a little easier to understand so now let's look at this one if you have polarizers 1 and 2 you will get the same result if you have 2 2 and 3 you will get a different result because the photon will pass through 2 but the photon will not pass for 3 similarly for 1 and this is 1 and 3, let's eliminate that 1 and 3 for 1 and 3, you will be different if you go down here by one until you are different from two and three you will get different one and three you get the same here one and two you get different two and three you get the same result one and three you get different one and two different two and three the same three different one and two different two and three different one and three the same one and two the same two and three different one and three different and this one and two equal two and three equal one and three equal here it will be that the photon does not pass through any of them so that it does not pass through one and does not pass through two, that is the same, so what have we seen here ?

In these two positions, the photon will definitely pass through both polarizers, whichever polarization you choose in these examples. a 1/3 probability that the photon will pass through the combination of polarizers that are chosen and the point of that long explanation is that it would be expected that if you did that experiment where you have a production of pairs of photons and then Alice and Bob randomly select between these polarizers and hold them in front of the photon and check if the photon passes or not, since you don't know what hidden variables those photons will have, you can at least say that you would expect that in 1/ On 3 of the occasions you would expect the results were the same, so if you did the experiment a hundred times, at least 33 times the experiment would come out the same and that's basically inequality of bells inequality of bait bells in this particular kay says to get the same result greater than or equal to one third of the times you do the experiment using the same logic we used for hats, gloves and scarves.

The only problem is that when you do the experiment you discover that you get the same result greater than 1/4 zero point 2 5 and that means that zero point 2 5 has to be greater than or equal to 1/3, which it clearly is not, so the bell inequality shows us that this hidden variable analysis cannot be true. There cannot be hidden variables because if there were, if there were absolutely determined variables that say whether or not the photon would pass through the polarizer, it would be expected that at least 1/3 of the time the result of the same measurements would be obtained, while experimentally. you only get that on the quarter of occasions and 1/4 is not greater than or equal to 1/3 and that seems to rule out the possibility of hidden variables so what does that leave?

Well, now that we said there were two possibilities. one was that in the mound the photons communicated with each other instantaneously and the other is that they had DNA embedded in them. If we discard the idea that they have embedded DNA or hidden variables, we are left with the only other conclusion, which is that somehow these two particles are in constant communication instantaneously, thus violating the principle of special relativity and what is now called the principle of locality, whether that is true or whether all this can be explained by another explanation not yet thought of is yet to be discovered.