Fundamental concepts in quantum physics: Bell's inequality - which was the basis of the Nobel Prize

Inspired by the winners of the 2022 Nobel Prize in Physics, we will discuss Bell's theorem without introducing formulas and inequalities. In this article we will illustrate how experiments with entangled particles contradict the assumption that there are hidden variables and that quantum mechanics is indeed reliable

In 1935, Einstein Podolsky and Rosen published a thought experiment that cast doubt on the integrity of quantum mechanics. In the published article it is claimed that if two particles come into contact with each other for a short time and then move away, an experiment can be conducted in which non-reciprocal physical quantities, for example position and momentum, will be accurately measured. The results of the thought experiment led the researchers to conclude that although quantum mechanics successfully predicts many experiments, it is not complete. In fact, the argument was that quantum mechanics is merely the opening shot towards the development of a theory that will faithfully describe nature and provide a more in-depth explanation of why momentum and position cannot be measured simultaneously. For three decades after the article was published, Einstein's doubts remained purely philosophical. On this, the Jewish physicist Otto Stern said, "There is no reason for us to break our heads anymore trying to understand whether there really are unknowable things and not the ancient question of how many angels can be answered on the tip of a needle", in an attempt to belittle Einstein's skepticism. All this changed in 1964 when physicist John Bell showed mathematically (in what is known today as Bell's theorem or Bell's inequality) that Einstein's view of the fragility of quantum mechanics was incorrect. Experiments carried out in laboratories around the world have shown time and time again that quantum mechanics is consistent and that the constraints that Einstein demanded of the universe are incorrect. In this article I will explain Bell's inequality that was experimentally broken by the 2022 Nobel Prize winners (without showing inequalities for their shares). I will assume that the readers know nothing about quantum mechanics. I will not mix words like superposition, the wave function, interlacing or spin (I will leave those at the end for the experienced reader). The next chapter will not even assume a basic introduction to classical physics. At the same time, I would emphasize that this is not an easy explanation, which can be found in abundance on the Internet. The concept behind this article is to explain Bell's idea in the simplest way, but not too simple so as not to disrupt the physics along the way.

the experimental system

Let's start by describing a device that contains several black boxes that are capable of performing several operations. We don't know exactly what is going on in the black boxes, but we know what will happen as soon as we turn them on, just like we know what will happen if we change the frequency on the radio without knowing the electromagnetic law. Although this device does not really exist, there is no reason why it cannot be built. I would emphasize that everything that will be explained here will describe an abstraction of experiments that did take place, and that the imaginary device succinctly describes a wide set of operations in complex experimental systems. The device contains three independent parts that are not connected to each other. When I say "not connected" or "independent" I mean that there is no physical connection between the three parts, i.e. there is no wire, pipe or cable connecting them, nor are there any forces they exert on each other. They can indeed be placed on the same table, but its effect will not be relevant to the experiment that will be detailed later.

What does the device consist of? Two of the three parts of the device are detectors. We will mark them with the foreign letters A and B. Each detector has a pointer that can be directed to one of three options, which we will mark with the numbers 1,2,3 (see video), and two lights, one green and one red. When the detector is on it will glow red or green. It doesn't matter how the detector determines which color to illuminate, what is important is that the device mediates information to the experimenter. The third part is a box placed between the two detectors (marked with the letter C) with a button on it. Every time the button is pressed, two particles moving in opposite directions will be emitted from the device towards the detectors. As soon as the particle hits the detector, one of the lights, and only one of them, will turn on. Since the parts of the device are not connected to each other, the bulbs can only turn on when the particle reaches them and not in any other way. The thought experiment we described will work a large number of times as follows: each of the detectors is randomly directed to one of the three numbers. Since there are two detectors and the pointer can be directed to one of the three numbers, the number of different experiments that can be conducted with this device is nine. We will mark them according to the numbers on which the hands are pointing, that is: (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1 ), (3,2), (3,3). For example (1,3) will describe an experiment in which the first detector was aimed at number 1 and the second detector at number 3. After that, we will press the button above box C, and two particles will be emitted. After a few moments the light in each of the detectors will turn on. Note that the light does not have to turn on simultaneously in both detectors. The detectors can be moved so that they are at different distances from each other and as a result the detectors will turn on at different times. By the way, the pointer can also be adjusted even after the particle has been released, but as long as it has not yet hit the detector. After both detectors have lit up, we will write down the result of the experiment. In our experiment notebook we will first write down the name of the experiment and its results using the letters G and R as an abbreviation for the green and red colors respectively. For example (1,3) GR will describe an experiment where the first detector was aimed at 1 and illuminated in the color green (Green) and the second detector was aimed at 3 and illuminated in the color red (Red). After a large number of repetitions, we will accumulate statistical information that will help us determine the probability that the two detectors light up in the same or a different color. This is similar to flipping a coin many times. If it is a fair coin, after a large number of trials, about half of the tosses came out a tree and the other half a fall. Of course, in the real world there are errors in measurements, but these errors will decrease as we run more and more experiments.

The results of the experiment:

The statistics accumulated from the measuring devices led to two interesting results:

1. In cases where the detectors were aimed at the same numbers (1,1), (2,2), (3,3) both detectors always illuminated the same color ie RR or GG.
2. In cases where the hands pointed to different numbers (1,2),(2,1),(3,1),(3,2),(1,3),(2,3), about a quarter of the cases the colors were the same and about three quarters of the cases The colors were different ie RG or GR.

We will first focus on the first experiment in which the two hands aimed at the same number - the natural question we would like to answer is why did the two detectors flash the same color in each experiment? Since the two detectors are not connected in any way, they cannot know which number the adjacent detector was targeting. At the same time, the experimental result can be easily explained when particles are taken into account. In relation to the number that the detector is calibrated to, eight measurement options can be assigned to the particle: RRR, RRG, RGR, GRR, GGR, GRG, RGG, GGG. That is, RGG will signify that for a particular particle emitted from the middle box (C), the measurement for it will be red if the pointer is pointed at 1, green if the pointer is pointed at 2 and green again if the pointer is pointed at 3. Note that the results of the experiment actually catalog eight different types of particles. Now let's go back to the first case: the fact that RG or GR is not possible when both hands point to the same number indicates that the two particles must be of the same type (try to convince yourself why this is necessarily true and that no other possibility exists). Since box C is not connected to detectors A and B, it cannot know in advance whether the hands are pointing to the same number or not and therefore we conclude that this feature also exists in the second case.

We will proceed to the second case when the two hands are pointing to different numbers - although the results of the experiment do not unambiguously determine the type of particle that box C will produce (because the experiment reveals partial information about the particle - only two of the three colors are detected by the two detectors), we can still reveal some interesting conclusions. Suppose the particle is of the RRG type, then only two of the six possible calibrations will cause both detectors to light up in the same color. These calibrations are of course: (1,2) and (2,1). The same is of course also true for the other types that mix the two colors (RGR, GRR, GGR, GRG). For RRR and GGG the detectors will light up in the same color 25% of the time. In total, if box C emits spontaneously and with equal probability all types of particles, the probability that the two detectors will light up in the same color when both are aimed at different numbers is a little more than a third. On the other hand, the probability measured by the measuring device is actually a quarter (XNUMX%)! We are left with a contradiction. The observational explanation for the first case does not agree with what we measured from the detectors in the second case. This is the essence of Bell's inequality and the contradiction between the prediction of "hidden variables" (the fact that the particles know in advance what the result of the experiment will be, as we have classified them) and what happens experimentally (which corresponds to quantum mechanics).

The quantum connection

In this chapter I will connect the parts of the device to the real world. From here on I will be forced to use professional terms to explain what is really happening in the experiment, at the same time I will continue to speak in general terms and will not go into details. Let's start with the detectors - their job is to measure the direction of the internal magnet of the particle. This magnet is sometimes referred to as spin. The main difference between a "quantum magnet" and a "classical" magnet (that is, between a particle and a macroscopic magnet), is that a "particle magnet" can only be found in two options depending on the calibration of the measuring device. When the spin is measured, a magnetic field is applied to the particle. This field, created in the vicinity of the detector, is associated with a north and south pole, similar to the diurnal magnets you know. When the quantum particle passes through the measuring device, its spin will be measured in the north or south direction of the detector and will be found only in one of these two options. On the other hand, a classical magnet can be in intermediate states, and in any orientation it chooses (not to be confused between superposition and an intermediate state which is a completely classical state with a general orientation). The number that the pointer indicates on the detectors describes the orientation of the measuring device. For example, the number 2 indicates that the device has rotated 120 degrees relative to the orientation set to number 1. Similarly, the number 3 represents a rotation of 240 degrees relative to 1. When the particle hits the measuring device, the color that will light up will indicate the direction of the spin, i.e. whether it is aligned with the magnetic field of the measuring device, or the opposite. When running the experiment, box C releases two particles with a spin similar to electrons, which are in an entangled state. When I write "intertwined state" I mean that during the measurement, the two particles immediately determine their state, or in other words, a measurement of one of the particles will determine in advance the result of the experiment on the other particle. In our case, every time one particle causes the measuring device to glow green, the other device will necessarily glow green (after the second particle hits it). Now it begs to be asked what will happen if the two measuring devices are not calibrated to the same orientation? Instead, let's start with a simpler question - what will happen to the second particle after the measurement of the first particle? It will be fixed on one of the turtles, but don't forget that the color (consider the spin up or down, or in direct or opposite orientation to the magnetic field of the detector) is necessarily associated with a number. That is, if the measurement of one particle determines the second particle to be in the 1G state for example, it will still be in a state of uncertainty regarding the measurement in the second detector if it aimed at a different number. Quantum theory teaches that this state is in a superposition of states from the measurement space of the other detector. That is, if the second detector is on number 2, G1 is the state equivalent to the superposition of the measurement modes in relation to the orientation of the second detector, which are 2G and 2R (to get green or red in the detector whose hand is pointed at 2). What determines the probability that the second detector will shine in one color or another is the orientation angle. In the case where the angle is 120 degrees, we reproduce the experimental results described above.

The article was mostly inspired His article of the physicist Nathaniel David Mermin.

Do you have a question or topic you would like me to write about? Contact me at noamphysics@gmail.com

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