**Background:**

Ever since quantum theory (QT) was invented almost 100 years ago (in 1925), it has led to a tremendous number of innovations that has revolutionized the world, including our lives and the business world. However, at the same time, from almost the very beginning, QT has created a number of deep mysteries that have shaken how we look at and understand the world, including making a great reassessment of whether we can actually explain what happens in nature, besides that we can no longer predict the outcome of every experiment, except to predict the probability of the outcome of experiments. It has led Einstein to make remarks like “Does the moon exist when there is no one looking at the moon?” and “God doesn’t play dice.”

In this summary paper, we recall these early mysteries, that led many physicists to acknowledge that although QT can make remarkable predictions (such as predicting accurately to one part to the billion), many also question whether QT is a complete theory, and its explanation of how and why things happen in nature may not be satisfactory. Some people thought that something like local hidden variable theory (LHVT) that states that there are also (hidden) variables that we are not aware of, and if we know and can specify the values of these hidden variables, then we will be able to predict the exact outcome of every experiment. Many physicists were even contemplating that LHVT may turn out to succeed QT and will be able to predict not only what will happen, but also to explain why certain things happen. Unfortunately, there didn’t seem to be any possible experiment that can be done to differentiate QT and LHVT.

This was the case for about 40 years until 1964 when the Irish physicist James S. Bell proved a remarkable but simple theorem (now known as Bell’s Theorem) that shows that QT and Local Hidden Variable Theories can lead to different experimental results, and therefore we can let experiments to decide on QT or LHVT. Starting in the 1970s and during the next half century, it has led to many remarkable experiments that repeatedly confirm all the predictions of QT and at the same time showed that Local Hidden Variable Theories are wrong. This not only declared that QT as the winner over LHVT, it also caused us to rethink how we should look at the world and whether there may be a limit in our understanding and description of the details of what is happening in an experiment. We will come back to Bell’s Theorem after we review the mysteries of Quantum Theory.

__Mystery of QT-I: Wave-Particle Duality:__

We now recall the series of quantum mysteries that started with the advent of QT. The first one was that the behavior of subatomic objects like electrons behave very differently from macroscopic objects, like baseballs, in the world around us. That is, in the microscopic world of electrons, they exhibit wave-like properties, but simultaneously also particle-like properties. The best explanation of this can be found in Feynman’s Lectures on Physics in his explanation of the double-slit experiment. [1] Physicists are all familiar with the double-slit experiments. For those who are not familiar with the double-slit experiments, it is best to read the description and explanation from Feynman himself in Ref. 1. A slightly shorter explanation can also be found in [2]. Note that in the explanation, we not only see the interference pattern of waves, but we also observe that the electrons arrive at the backdrop with full clicks, but never a half click, indicating that electrons also behave like particles even though simultaneously they are also behaving like waves. This phenomenom is known as wave-particle duality of QT, i.e., subatomic particles display both characteristics, showing behavior like a wave, and also showing behavior like a particle.

__Mystery of QT-II: The Act of Observance Can Change What You Are Observing:__

If we follow the discussion of the double-slit experiment by Feynman, and if we try to determine which of the slits that the electron goes through, we learned that the act of observance actually changes the result of the experiment. This is actually perfectly understandable in the microscopic world, because in order to observe the path of the electron, for example by placing a light after the slits. Since electric charges scatter light, then noticing whether the light is coming from near slit 1 or slit 2, we will be able to determine whether the electron went through slit 1 or slit 2. However, when we do that, the resulting pattern on the backdrop changes drastically so that the interference pattern we saw in the original double-slit experiment signaling the wave-like property of electrons has disappeared, and replaced by the pattern characteristic of particle-like behavior. In the macroscopic world, we don’t see this because the disturbance from the act of observing is not large enough to change the result of the experiment.

__Mystery of QT-III: One Cannot Simultaneously Measure Precisely the Position and Momentum of any Object (also known as Heisenberg’s Uncertainty Principle):__

One way of understanding this principle is that in order to measure the precise position of an object, you have to use shorter and shorter wavelength of light. But the shorter the light’s wavelength, the higher is the frequency of the light, since the light’s frequency is in inverse proportion to the light’s wavelength. And the higher the frequency of the light, the more energy is the light, since E=hv, where E is the energy, v is the frequency, and h is Planck’s constant. That is why to measure precisely the position of an object, you need to use shorter and shorter wavelength, or higher and higher frequency, and therefore you impart more and more energy to the object. If the object is a microscopic object like an electron, the energy you impart from the observance of its position would disturb its velocity greatly, thus resulting in Heisenberg’s Uncertainty Principle.

__Mystery of QT-IV: In the Subatomic World, We Can Only Predict the Probability Distribution of Certain Physical Happenings in the Future:__

In classical physics, if we are given the initial conditions of an object, then using the laws of physics, we can predict precisely the future behavior of an object. However, in the subatomic world, because particles also have wave-like characteristics, we can no longer predict precisely the behavior of an object even if given the initial conditions.

Instead, in QT, the state of an object is described by a complex wave function C (complex in the mathematical sense of real versus complex numbers). The probability of finding the object in a certain state and time is given by the square of the complex function C, i.e., |C|^{2}. In QT, we can only predict the probability of finding the object in a certain position and time, but never precisely that the object will be in a certain position at a particular time. In other words, we change the precise prediction of classical physics to the probability prediction of quantum theory.

This led many people, including Einstein, to make the remark that God does not play dice and question whether there is a more fundamental theory than quantum theory so that the uncertainties can be removed and the theory can then be deterministic, and not probabilistic. That was what led many people to the LHVT as mentioned in the Background section at the beginning of this article.

__Local Hidden Variable Theory (LHVT) and Bell’s Theorem:__

Before we discuss the experiments done in the past half century related to Bell’s Theorem and the implications of these experiments, we first discuss two classic thought experiments from 1935.

** Schrodinger’s Cat:** As we discussed earlier that the act of observing can change what we are observing. So in the double-slit experiment, if we try to detect which slit the electron went through, the interference pattern that was observed indicating the wave-like properties of electrons disappeared. Within the wave function mathematical description of QT, the wave function was originally a superposition of states, but then the act of observation causes the wave function to collapse to a specific state.

This led Schrodinger to the following thought experiment in 1935. There is a radioactive atom inside a box that is connected to a radioactive detector that is connected to a hammer that breaks a glass jar which then releases a poison gas, and inside this box is a cat. The radioactive atom has certain probability of radioactive decay, e.g., with a half-life of about 10 minutes. From QT, the radioactive atom is described by a wave function that is a superposition of two states, a decay state and a non-decay state. A decay state will release the poison that will kill the cat. A non-decay state will not release the poison and the cat remains alive. Before the box is opened when we don’t know what is inside the box, the quantum wave function of what is inside the box is a superposition of a live cat and a dead cat. However, when the box is opened, we will find either a live cat or a dead cat, and never a half-dead cat and a half-alive cat. In other words, the act of observation caused the wave function to collapse from a superposition of states to a specific state. This is analogous to the double slit experiment with electrons that the act of observing changes what we are observing.

** The Einstein-Podolsky-Rosen (EPR) Paradox: ** Einstein together with Boris Podolsky and Nathan Rosen proposed another thought experiment, also in 1935. This involved a QT system at rest and zero angular momentum, also known as spin-0, that emits two photons in the opposite directions. Since photons also have spins (their spins are either up or down), and linear momentum and angular momentum are conserved, if one photon is moving to the left and is measured to have spin up, then the other photon is moving to the right and if measured must then have spin down. If we provide sufficient time for these two photons to travel a large distance, then if we measure the spin of the left photon and find it to be spin up, then if we measure the spin of the right photon, then its spin has to be down. This seems to say that the photons seem to be entangled, i.e, measuring the spin of the photon going to the left will automatically tell you the spin of the photon going to the right. This is known as quantum entanglement. Since the distance between these two photons can be sufficiently large so that no information can be transmitted from the left photon to the right photon, unless we are willing to give up one of the cornerstones of the theory of special relativity that stipulates that no information can be transmitted faster than the speed of light. This led Einstein to make the remark that QT seems to allow “spooky action at a distance.” This thought experiment of Einstein/Podolsky/Rosen is known as the EPR Paradox. Now we are ready to discuss Bell’s Theorem.

** Bell’s Theorem:** In 1964, the Scottish Physicist James S. Bell proved a very remarkable and extremely important theorem (some people have even proclaimed that it is the most profound discovery of science) that showed QT and LHVT do not always make the same predictions.

__In other words, we can now do experiments to decide whether QT or LHVT is correct or wrong (or both could be wrong).__Although extremely important, Bell’s Theorem was actually relatively simple to prove in terms of the mathematics used. The original proof by Bell [3] makes ingenious use of logic but only high school mathematics; other proofs developed by others later all make use of logic and also did not require the use of sophisticated mathematics. [4] However, the proof does not require to know the detailed specification of the hidden variable theory, but it does require the use of locality, i.e., no information can be transmitted faster than the speed of light. That is why LHVT in Bell’s Theorem applies to all hidden variable theories as long as it is local. Bell’s Theorem can be shown as an inequality and is therefore also known as Bell’s Inequality. Although the proof is not long or complicated, we are not including the proof in order to make this article to be of reasonable length, but the proof can be found in, e.g., [4] and [5].

The first people to use Bell’s Theorem to compare the predictions of QT and LHVT were S.J. Freedman and J. F. Clauser in their classic paper in 1972. [6] The experiment is similar to the setup in the EPR paradox except that photons with spin up or down were replaced by photons with polarization which could have two values – up or down. When the results at the photon detectors were compared, the ups and downs did not match the ways predicted by LHVT, i.e., Bell’s inequality was violated, showing that LHVT is wrong. However, the experimental results were completely consistent with the predictions of QT. During the last half century, the result of this experiment has been confirmed by many other experiments, including developing more sophisticated experiments that removed some of the possible ambiguities or so-called loop holes in the Freedman-Clauser experiment This has removed a lot of skeptics of QT, and more and more people think that QT may be a correct theory, in spite of the fact that its many mysteries seem to contradict our understandings and explanations of how nature behaves. As a matter of fact, we may even have to accept the fact that Is how nature behaves, or perhaps until another great breakthrough in the future that could possibly provide a better explanation of what we observe. We now discuss refinements of the Freedman-Clauser experiment.

** Refinement and Removing Possible Loopholes in Experiments on Bell’s Theorem:** When they were doing their 1972 experiment, Friedman and Clauser were already aware of possible “loopholes” in their experiment. For example, could it be possible that their laboratory instrument might have been leaking information to each other, after all, the correlated information of the separated photons only showed up after the measurements of the separated photons. In 1982, Alain Aspect in France refined the experiment by switching the direction of the photons’ polarization every 10 nanoseconds when the photons were already in the air and too fast for the two pieces of laboratory instrument at the two ends to communicate with each other. But results turned out to be the same, i.e. experimentally LHVT is again wrong and QT is correct. [7]

Another possible loophole was that the polarization directions in Aspect’s experiment in France had been changed in a regular and therefore theoretically predictable fashion that could be sensed by the photons or detecting instruments. Then Anton Zeilinger of Austria in 1992 used random number generators to change the direction of the polarization measurements while the photons were in flight. The experimental results again confirmed that LHVT is wrong and QT is correct. [8]

This kind of experiments has been done many times in the last several decades, and the winner has always been QT.

By now, scientists have done experiments with entangled particles, and entanglement is accepted as a main feature of QT and is being put to work in cryptology and quantum computing. An application in cryptology is to send messages using entangled pairs, which can send cryptographic keys in a secure manner, because any eavesdropping will destroy the entanglement, thus alerting the people involved in the transmission that the message has been disturbed. In 2017, Dr. Zeilinger used this technique with entangled photons beamed from a Chinese satellite called Micius to have an encrypted 75-minute video conversation with the Chinese Academy of Sciences’ Dr. Jian-Wei Pan, one of his former students. [9]

__Summary:__

Quantum Theory started a revolution in physics, as well as in our business world and our everyday lives. It changed our understanding of nature and our description of the microscopic world at the scale of atoms. It introduced many unfamiliar concepts, such as particle-wave duality, uncertainty principle, the act of observance can change what one is observing, and the probability interpretation in place of absolute predictability. These unfamiliar concepts led many people, including physicists, to question whether QT can be a true physical theory, in spite of its huge number of experimental predictions, and many physicists thought that there are hidden variables that we are not familiar with. Once we can specify these hidden variables, then we should be able to remove many of these mysteries.

But in 1964 James S. Bell proved a very simple theorem that shows all local hidden variable theories (LHVT) cannot make the same experimental predictions as QT. Experiments in the last half century have repeatedly demonstrated that LHVT is wrong, and QT is correct. This reduced significantly the sceptism toward QT, but at the same time, it has introduced mind-boggling concepts and questions such as quantum entanglements, spooky action at a distance, does the moon exist when there is no one looking at the moon. It causes us to wonder what is reality, can we describe or explain reality, or perhaps we are only able to describe the results of experiments, but not always able to describe what went on during the experiments.

We are definitely entering a challenging and exciting time in physics, hoping and waiting for the next breakthrough. There may not be any more holy quails, including perhaps locality.

This year’s Nobel Prize in Physics were awarded in October 2022 to A. Aspect, J. F. Clauser, and A. Zeilinger.

Note: Stuart Freedman passed away in 2012, and is therefore not eligible for the Nobel Prize. James S. Bell also passed away in 1990, and was also not eligible for any Nobel Prize after 1990.

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[1] R. P. Feynman, R. B. Leighton, M. Sands, *The Feynman Lectures on Physics, Volume 3: Quantum Mechanics*, Addison-Wesley Publishing Company Inc., Palo Alto, 1965. The double-slit experiment is eloquently explained in Chapter 1 “Quantum Behavior.”

[2] A slightly shorter explanation of Ref. 1 can also be found in the article “Wonders and Mysteries of Quantum Physics”: https://www.dontow.com/2020/09/wonders-and-mysteries-of-quantum-physics/.

[3] J. S. Bell, *Physics *1, 195 (1964).

[4] See, for example, the video by Alvin Ash “The EPR Paradox and Bell’s Inequality explained Simply”: https://www.youtube.com/watch?v=f72whGQ31Wg.

[5] “Paradoxes of Quantum Physics, Bell’s Theorem, and What Do Experiments Tell Us”: https://www.dontow.com/2020/12/paradoxes-of-quantum-physics-bells-theorem-and-experimental-confirmation/.

[6] S. J. Freedman and J. F. Clauser, “Experimental Test of Local Hidden-Variable Theories,” *Phys. Rev. Lett.* **28**, 938, 3 April 1972. This was Freedman’s Ph.D. thesis at the University of California at Berkeley, and Clauser was a post-doctoral fellow at the University of California at Berkeley.

[7] A. Aspect, J. Dalibard, G. Roger, *Phys. Rev. Lett.* **49**, 1804, 1982.

[8] G. Weihs, T. Jennewein, C, Simon, H. Weinfurter, A. Zeilinger, *Phys. Rev. Lett.* **81**, 5039, 1998.

[9] J. Yin, et. al, Nature, **582**, 501, 2020.