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Our previous article “Wonders and Mysteries of Quantum Physics” discusses how Quantum Physics (QP) [1] completely revolutionized our industrial world and our daily lives since its discovery about 100 years ago. Everyday we utilize a variety of products based on Quantum Physics. [2] That article also pointed out that QP introduced many mysteries, such as particle-wave duality, the act of observation can change what we are observing, uncertainty principle, our physical laws can only give us a probabilistic, and not a deterministic, prediction of the future.

These mysteries, especially the probabilistic interpretation, or a superposition of states, led many people to question from the beginning of QP around the mid 1920s whether there is a more fundamental theory than QT that would lead to a deterministic prediction. The most famous critic was Albert Einstein, who made critical comments such as “does the moon exist even when no one is looking at it” and “God doesn’t play dice.” Many people thought that there are probably physical variables that we are not aware of. Because these variables could have different values, and if we can determine their values, then we would have a deterministic prediction.

These are known as “hidden variable” theories. Even though the usefulness of QT became more and more apparent as more and more products based on QT permeated our lives, this debate never went away, partially because no one could think of any experiment that could be done to differentiate the predictions of QT and the predictions of hidden variable theories.

That ended in 1964 when the Irish physicist James S. Bell proved a remarkable but simple theorem (now known as Bell’s Theorem) that shows that Quantum Theory and local hidden variable theories can lead to different experimental results. [3] Therefore, this is no longer an academic debate, but a debate that can be decided by experiments, which is the fundamental concept behind physics. Before we discuss Bell’s Theorem and the subsequent experimental results, we need to make a digression to discuss two precursors of Bell’s Theorem.

Precursors that led to Bell’s Theorem – Schrodinger’s Cat:

As we discussed in my previous article about the double slit experiment, the act of observing can change what we are observing. So in the double slit experiment with the electrons, when we insert a light source behind the double slits so we can determine which slit the electrons went through, the interference pattern observed downstream is disturbed and the interference pattern is no longer seen. Note that this does not mean that the electrons did not go through both slits (after all, the electrons already went through the slits before reaching the position of the light source), but the light from the light source did disturb the motion of the electrons from that point forward, and as a result the interference patterns have been changed. Therefore, the act of observing can change what we are observing. 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 is also the reason why in the double slit experiment when we detect the electrons in the backstop, we always hear a full click and detect a whole electron, and never hear a half click and detect half an electron. In other words, in the act of detection, the electron’s wave function collapses from a superposition of states to a single state.

This led to the thought experiment of Schrodiner’s cat proposed by Schrodinger in 1935. [4] In this thought experiment 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., for certain atoms 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.

This leads to a deep philosophical or logic question of what physics is supposed to be able to achieve? Is the ultimate objective of physics to be able to explain why certain things happen besides predicting what will happen? Or is the ultimate objective of physics to be able to predict only what we can observe, i.e., only the outcome of experiments? This led Einstein to make the comment “Does the moon exist when there is no one looking at the moon?” I think before QT, most physicists would think that it is the former, i.e., physics should be able to explain why certain things happen as well as predicting what will happen. With the advent of QT, physicists became divided on what is the ultimate objective of physics.

Another Precursor to Bell’s Theorem: Einstein-Podolsky-Rosen Paradox:

Also in 1935, Einstein, together with two colleagues Boris Podolsky and Nathan Rosen at the Institute at Advanced Study at Princeton, proposed another thought experiment. This involved a QM 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. However 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 that transmission occurs essentially instantaneously or at least faster than the speed of light. Therefore, 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 (or allow what Einstein called “spooky action at a distance”), then it appears that QT cannot be right or complete. This is known as the “EPR Paradox.”

However, Niels Bohr [5] argued that information was never transmitted from the left photon to the right photon. The two photons were entangled particles in a quantum superposition state, and that was the state of the system at that time. When we measure the spin of the left photon, we disturb the system, and the system’s wave function collapes from a superposition of states to a specific state, and that was the state of the system at that later time. In Bohr’s explanation, the wave function provides a method to calculate the outcome of experiments (i.e., if you do the experiments many times, you will find that half of the time the cat is alive and half of the time the cat is dead), but it does not represent the physical state of the system (i.e., the cat is half alive and half dead).

Bell’s Theorem:

This debate on the completeness and accuracy of QT continued, with no agreement and definitely no consensus. Although people thought that QT provides an extremely accurate description of experimental results, many people thought that QT is not a complete theory, and its explanation of how and why things happen in nature is not satisfactory. Some people thought that something like local hidden variable theory (LHV) 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 LHV.

This came to an end in 1964 when the Irish physicist John S. Bell prove a very simple but extremely important theorem that tells us that QT and LHV do not make the same predictions (some people have even proclaimed that Bell’s Theorem is “the most profound discovery of science”). In spite of its importance, it turns out to be a simple theorem to prove using nothing more than logic and high school mathematics, although the proof does require ingenuity.

Bell’s Theorem states that QT and LHV do not always make the same experimental predictions. To prove this theorem, it is sufficient to show one experiment that will lead to different predictions for the two theories. We now present a proof developed by others that is simpler than Bell’s original proof. The proof makes use of logic and some simple concepts inherent in all “local hidden variable theories.” This theorem will show that QT and LHV will not always have the same predictions.

For people who are not interested in the mathematical proof of Bell’s Theorem and the quantitative predictions of QT for the experiments discussed in Bell’s Theorem, they can skip the rest of this section and continue with the section on Experimental Results.

The experiment is the one discussed in the EPR paradox: a QM system of zero angular momentum (also known as spin-0) that emits two photons in the opposite directions. As discussed in the earlier section on the EPR paradox, because of the conservation of linear momentum and angular momentum, if one photon travels to the left and has spin up, then the other photon must travel to the right and have spin down.

Proof of Bell’s Theorem for Local Hidden Variable Theories:

Many people have discussed this proof. I will use the proof as presented in the video by Alvin Ash. [6][7]

In this experiment, when we measure the spin of the photons, we can choose to measure its spin relative to any direction. Independent of the direction, its spin is either spin up or spin down relative to that direction. In any LHV theory because of the hidden variables which are not defined in QT, the left and right photons are always a pair of (left up, right down) or a pair of (left down, right up). This is different than in QT, where each of the photons in the pair is a superposition of an up-state and a down-state. The uncertainties or the superposition of states as in QT is removed by the very definition of hidden variable theory. Note that we also used locality in the sense that no information can be transmitted instantaneously or at least not faster than the speed of light, that is, the information on measuring the spin of the left photon when the two photons have moved apart to a large enough distance cannot be transmitted to influence the measurement of the right photon. Separating the left measuring apparatus and the right measuring apparatus at far enough distance will guarantee that.

Now, let’s consider two people doing the measurements: Alice doing the measurement of the left photon, and Bob doing the measurement of the right photon. Part of Bell’s ingenuity is to point out the need to do this experiment at least three times: (1) Measurement relative to the z-axis, (2) measurement relative to the x-axis which is 90 degrees rotated from the z-axis, and (3) measurement relative to the q-axis at some angle, say 45 degrees, between the z-axis and the x-axis. Let’s denote a spin-up measurement along the z-axis as Z+, and spin-down measurement along the z-axis as Z-. Similarly, let’s denote a spin-up measurement along the x-axis as X+, and a spin-down measurement along the x-axis ad X-. Finally, let’s denote a spin-up measurement along the q-axis as Q+, and a spin-down measurement along the q-axis as Q-. So there are eight possible results for the 3 sets of measurements of Alice along the z-axis, x-axis, and q-axis. The results are:

E1:  Alice:  Z+, X+, Q+

E2:  Alice:  Z+, X+, Q-

E3:  Alice:  Z+, X-,  Q+

E4:  Alice:  Z+, X-,  Q-

E5:  Alice:  Z-,  X+, Q+

E6:  Alice:  Z-,  X+, Q-

E7:  Alice:  Z-,  X-,  Q+

E8:  Alice:  Z-,  X-,  Q-

Let P (Z+, X+) denote the probability that such an experiment would result in Alice measuring the left photon with a positive spin in the z-axis and in Bob measuring the right photon with a positive spin in the x-axis (which means that Alice would measure the left photon with a negative spin in the x-axis). Then looking at the 8 possible measurements of Alice for the left photon, only E3 and E4 would result such a combination. This gives the following equation:

Eq. 1: P (Z+, X+) = (E3 + E4)/8 = 2/8 = 0.25

Similarly, let P (Z+, Q+) denote the probability that such an experiment would result in Alice measuring the left photon with a positive spin in the z-axis and in Bob measuring the right photon with a positive spin in the q-axis (which means that Alice would measure the left photon with a negative spin in the q-axis. Looking at the 8 possible measurements for Alice of the left photon, only E2 and E4 would match such a combination. This gives the following equation:

Eq. 2: P (Z+, Q+) = (E2 + E4)/8 = 2/8 = 0.25

Finally, let P (Q+, X+) denote the probability that such an experiment would result in Alice measuring the left photon with a positive spin in the q-axis and Bob measuring the right photon with a positive spin in the x-axis (which means that Alice would measure the left photon with a negative spin in the x-axis. Looking again at the 8 possible measurements for Alice of the left photon, only E3 and E7 would match such a combination. This gives the following equation:

Eq. 3: P (Q+, X+) = (E3 + E7)/8 = 2/8 = 0.25

We will shortly come back to Eqs. (1), (2), and (3).

Now we discuss another ingenious observation of Bell. Since P (Z+, X+), P (Z+, Q+), and P (Q+, X+) are probabilities; each one is either positive or zero. So we can write the following inequality:

Eq. 4: (E3 + E4)/8 ≤ (E3 + E4 + E2 + E7)/8 = (E2 + E4)/8 + (E3 + E7)/8 = 4/8 = 0.5

which can be rewritten as:

Eq. 5:   P (Z+, X+) ≤ P (Z+, Q+) + P (Q+, X+), which is known as Bell’s Theorem

Because it is expressed as an inequality, it is also known as Bell’s Inequality. Since all the probabilities in Eq. 5 can be measured in an experiment, Bell’s Inequality can be tested.

Just to be sure, we can also substitute Eqs. 1-3 into Eq. 5, and get 0.25 ≤ (0.25 + 0.25) = 0.5, which is satisfied, which shows that Local Hidden Variable Theories satisfy Bell’s Inequality.  But whether it agrees with experiments is another matter to be seen.

Predictions of Quantum Theory:

In the above experiment, all the measurements can be predicted by QT. We know that if Alice measures the left photon and found its spin along the z-axis to be positive, then when Bob measures the right photon, he will find its spin along the z-axis to be down. But what if Bob measures the spin of the right photon along the q-axis, say at 45 degrees relative to the z-axis or x-axis, what is the probability that he will find the spin of the right photon to be positive. QT can calculate this probability and gives the result:

Eq. 6: P (Z+, Q+) = [sin (45°/2)]².  [8]

Similarly, the QT predictions for

Eq. 7: P (Z+, X+) = [sin (90°/2)]²

Eq. 8: P (Q+, X+) = [sin (45°/2)]²

We can now write the QT predictions for the left side and right side of Eq. 5 of Bell’s Theorem.

Left Side = [sin (90°/2)]² = 0.5.

Right Side = {[sin (45°/2)]² + [sin (45°/2)]²} = 0.293.

This gives:

Eq. 10: Left Side = 0.5, and Right Side = 0.293, i.e.,

the Left Side is not ≤ the Right Side.

Therefore, Bell’s Inequality of Eq. 5 for local hidden variable theories is not satisfied in QT. This means that QT and LHV do not always give the same predictions.

Experimental Results:

Although it took about a couple of years for people to realize the significance of what Bell had done, once it was understood, it shook up the whole physics community. Starting in early 1970, many experiments have been done during the next 40+ years. The first experiment was completed in 1972 by Stuart Freedman and John Clauser at the University of California at Berkeley. [9]  Similar and more refined experiments have been done by several other groups in different parts of the world in the next three-four decades. These experiments have all confirmed the predictions of QT, and they showed that the Bell Inequality, which is required by all local hidden variable theories, is violated. Furthermore, there are variations in these experiments, such as in the choice of particles, e.g., photons, electrons, as well as other particles, their results are all consistent with the predictions of QT, and not consistent with the predictions of LHV.

Therefore, the view toward QT for the last quarter century has shifted somewhat from that during the three decades between 1935 and the mid 1960s. Although physicists are still not completely satisfied with QT, especially in its explanation of what is happening in nature, and not just on whether it can predict accurately the results of measurements, most physicists no longer have their hopes on local hidden variable theories.

Besides experiments related to Bell’s Theorem, QT during the last 100 years has led to a huge number of extremely innovative and useful products that have revolutionized and permeated our industry and our everyday life. QT has also made many very detailed predictions that are in agreement with experiments, some remarkably to 10 decimal places. Therefore, fewer physicists now question the predictions of QT. However, because there are still mysteries of QT and we would like to be able to explain what happened in an experiment, and not just what will be the predictions of experimental measurements, many physicists still believe that there may be a more fundamental theory that on the one hand can duplicate the predictions of QT, and on the other end also resolve at least some of the mysteries of QT and provide a more satisfying description and explanation of nature.

Quantum Entanglement and Quantum Computing:

When we construct computers making full use of QP, the mystery of quantum entanglement has important implications for encryption and computer security.   Quantum entanglement means that two physical entities that originated from the same entity are correlated no matter how far apart they are.  So once you make a measurement on one entity and cause the wave function to collapse to a specific state from a superposition of states, then the system is disturbed and the wave function for the other entity also collapses from a superposition of states to a specific state.  Therefore, if the two entities contain an encrypted message, and the message is intercepted by an intruder, then you know that the message has been disturbed.

Quantum computers can also significantly increase the processing power and speed of computers.  Besides providing the ability to solve many problems which are currently not solvable from a practical point of view, such powerful and faster computers can also break many currently used security algorithms.  Some of the implications from QP for computers and information networking will be discussed in the next release of this website.

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[1] Quantum Physics (QP) is also known as Quantum Mechanics (QM) or Quantum Theory (QT).

[2] E.g., PC, cell phone, TV, radio, GPS, light bulbs, electron microscopes, x-rays and medical imaging, electronic appliances, digital cameras, lasers, Internet, modern aircrafts, satellites, nuclear powers, missiles, …

[3] Bell’s Theorem applies only to hidden variable theories that are local, i.e., no information can be transmitted faster than the speed of light, as required by Einstein’s theory of special relativity.

[4] Erwin Schrodinger was an Austrian physicist who with the German physicist Werner Heisenberg were the co-inventors of QM. Heisenberg formulated QM using a matrix formulation, and Schrodinger formulated QM using a wave equation formulation. It turned out that the two formulations are equivalent. Heisenberg received the 1932 Nobel Physics Prize, and Schrodinger received the 1933 Nobel Physics Prize.

[5] Niels Bohr is the famous Danish physicist who invented the Bohr model of the atom to provide a theory of the atoms, and also made many great contributions to the formulation of QT. He won the Nobel Prize in Physics in 1922.

[6] Although his training was not in physics, Alvin Ash is extremely good in explaining complex physics topics to laymen in easy-to-understand videos. My discussion on the proof of Bell’s Theorem is based on his video “The EPR Paradox and Bell’s Inequality explained Simply”: https://www.youtube.com/watch?v=f72whGQ31Wg. Note: Because Bell’s Theorem can be shown as an inequality, Bell’s Theorem is also often known as Bell’s Inequality.

[7] There are many documentary lectures and videos on Quantum entanglement and Bell’s Theorem.  A good and enlightening documentary is the video by Jim Al-Khalili “The Secrets of Quantum Physics with Jim Al-Khalili (Part 1/2):  Spark”: https://www.youtube.com/watch?v=ISdBAf-ysI0.  Another excellent program is the more recent (2019) PBS Nova program “Einstein’s Quantum Riddle”:  https://www.pbs.org/wgbh/nova/video/einsteins-quantum-riddle/.  This program provides an excellent summary of the Einstein-Podolsky-Rosen Paradox, Quantum Entanglement, and Bell’s Theorem.  It also reports on a very recently completed experiment that is a refinement of the Freedman-Clauser experiment (see Ref. 9), and once again verifying that Quantum Theory is correct.

[8] There is a proportionality constant in front of the right side of Eq. 6, Eq. 7, and Eq. 8. Since it is the same proportionality constant that will appear on both sides of the equation, we have ignored the proportionality constant.

[9] S. J. Freedman and J. f. Clauser “Experimental Test of Local Hidden-Variable Theories,” Phys. Rev. Lett. 28, 938, 3 April 1972. This was also the Ph.D. thesis of Freedman, who was a student of Professor Eugene Commins, but the idea of doing this experiment came from Clauser, who was a postdoctoral fellow under Professor Charles Townes, the inventor of maser/laser and the 1964 Nobel Physics Laureate who just came to UCB in 1967. The experiment made use of some experimental apparatus from an earlier experiment by Commins. So this project was jointly sponsored and funded under Commins and Townes. Freedman and I were undergraduate and graduate students at UCB in the same period.