 Quantum state superpositions and entanglement are two of the most fundamental concepts in quantum mechanics and also two of its most misunderstood, and they are turning out to be the key to the next generation of quantum computing. In our first chapter, the microscopic, we covered the double-slit experiment that showed how photons and electrons display both wave and particle properties. It's called wave particle duality or complementarity. The key to the experiment was to observe what happens when we detect which of the slits a particle went through. For photons, we never explained how we could detect a photon without disturbing its path. So this final chapter brings us full circle, where we will cover in detail how this was done. In our second chapter on the atom, we covered Schrodinger's equation with its probability wave, Eisenberg's uncertainty principle and Pauli's exclusion principle with electron spin. These constitute the base physics for understanding superpositions and entanglement. We'll cover exactly what quantum superposition and entanglement are. We'll cover Einstein's problem with quantum mechanics and his prediction that we will someday find hidden variables to explain entanglement. We'll cover a thought experiment designed to show that hidden variables cannot exist. It's called Bell's theorem or Bell's inequality. We'll cover a real experiment that uses entangled photons to create ghost images that produce a bell inequality. Along the way, we'll clear up a few misconceptions about Schrodinger's cat and the quantum eraser. We'll end with a look at quantum computing and how it directly manifests and leverages these quantum properties. Our first encounter with quantum superpositions will be the double slit experiment, so in preparation we'll cover some key characteristics of light polarization. We understand light as an electromagnetic wave. The direction of the electric field is called the wave's linear polarity. Here we see the polarity at different angles from a fixed reference. It is also possible for the polarity to be rotating clockwise or counterclockwise around the line of motion. These properties hold for the basic unit of light, the photon. When light passes through a polarized lens, the amount of light that makes it through depends entirely on the angle between the incoming light's polarization and the polarization direction of the lens. To see this, here are a couple of experiments you can do at home if you have three pairs of polarized glasses. Photons leaving the background table have a wide variety of polarizations. We start with the lens that only allows light polarization in the vertical direction to pass through. All the other light is blocked. We'll call this lens A. If we bring a second lens, lens C, and orient it the same as the first, all the light passes through A passes through C. But as we rotate lens C, we see that the amount of light passing through is going down. By the time we reach 90 degrees, C is blocking all the light that passes through A. Now if we bring in a third lens, lens B, and place it in between the first two and angle it at 45 degrees, we see that light that could not make it through C before is now coming through. In other words, lens B, designed to reduce the amount of light that reaches C, actually enables more light to get through C. To see what is happening here, we need to go down to the photon level. Classically, we calculated the percentage of light that goes through a lens, but a photon will go through or not go through. It cannot be divided. In quantum mechanics, it's the angle between the orientation of the photon's quantum state and the orientation of the lens's polarization that provides the probability for passing through the lens. In addition, the interaction between the lens and the photon will change the orientation of the photon's state to equal the orientation of the lens it passed through. With this understanding, we can examine how light made it through lens C once we added lens B. Here we have a number of photons with random polarizations trying to pass through the vertically polarized lens A. Some make it and some don't. All the photons that passed through A have now been changed to have the quantum state vertical to match the lens. With this polarization, the probability of passing through lens C, which is rotated 90 degrees from the vertical, is zero. No light gets through C. Now we introduce lens B, which is rotated 45 degrees from vertical. We see that some of the vertically polarized photons coming through lens A will pass through lens B. In addition, the interaction between the photons and lens B change the photon's quantum state to oriented at 45 degrees to match the lens. This enables some of the photons that passed through lens B to now pass through lens C. The key takeaway here is that objects like lenses, crystals, electric fields, etc. can and do modify the quantum states of the particles that encounter them. Based on the wave nature of particles, superposition is the combining of multiple waves. For example, here we see two waves with amplitudes A and B. When they combine, the superposition state has an amplitude of A plus B. Their relationship is a linear. In this next example, where one has an amplitude A and the other has an amplitude minus A, the superposition is zero. They cancel each other out. Remembering that a physical system can be described by a wave function and Schrodinger's wave equation, their quantum states can be linearly combined like these waves. This is the principle of quantum linear superposition. The double slit experiment with photons helps illustrate how this linear superposition works. As light flows through the process, we'll keep track of the quantum state of the photons. We start out with light being passed through a linear polarizer. On exiting the polarizer, we mark the first quantum state as zero for location and V for vertically polarized. As it travels to the double slit, it evolves into a linear superposition state for S1 and S2. It represents the state where it could be at either S1 or S2. For photons reaching the screen from S1, state evolves into one that includes coefficient amplitudes that vary for different screen locations. The same is true for photons reaching the screen from S2. Only the amplitudes will be different. And unique to quantum mechanics, photons reaching the screen from the S1 plus S2 state evolve to a linear superposition of the two, like a wave passing through both. We square the wave functions to get the probabilities. We see that the probability of hitting any particular point on the screen has four components. One is for photons going through S1, one is for photons going through S2, and two are for the photons going through both. It is the interaction between these two that come from the superposition states on the far side of the double slit that create the interference pattern. To find out which way a photon went, two quarter-wave plates are placed in front of the slits. A quarter-wave plate is a special crystal that can change linearly polarized light into circularly polarized light. Plate 1 in front of slit 1 will change the photon's polarization to be clockwise, while plate 2 in front of slit 2 will change it to be counterclockwise. These are reflected in the photon's new quantum state where R is for clockwise and L is for counterclockwise. Once the photon reaches the screen, we can measure its polarization and know which slit it went through. But because the left and right polarization terms are orthogonal, they cancel out when we calculate the probability distribution. We are left with a probability distribution that only contains terms for the two slits giving us the blob instead of the interference pattern. Now if we remove the quarter-wave plates, we get back to superposition states and the interference pattern. In the early days of quantum mechanics, some physicists proposed that linear superposition was appropriate for macroscopic objects, and that superposition states only degenerate into base states when the system is observed or measured, implying the need for a human. To counter these misconceptions, Schrödinger, with a bit of humor, proposed a thought experiment now called Schrödinger's cat. It went like this. Suppose we had a cat penned up in a box with a tiny bit of radioactive substance, so small that in the course of an hour one of the atoms might decay, but with an equal probability that it does not decay. He added a Geiger counter to detect the decay should it happen. The Geiger counter is hooked up to a lever that drops a weight on a glass bottle of hydrocyanic acid should it detect a decay. The released poison gas kills the cat. If we were to consider the quantum state of the cat during this hour, we'd say it is in a superposition of alive and dead. And this state would persist until we opened the box and the subjective observer-induced collapse of the wave function revealed at the state of the cat, alive or dead. First, the idea that life and death could be considered quantum states isn't right. And second, the idea that the cat, if found dead, died when the box was opened is ridiculous. An autopsy would prove that it had died earlier than that. The real situation has the decaying atom in a linear superposition state of decayed plus not decayed. Its wave function collapses at decay time when the Geiger counter encounters it. The subsequent observation by a human records only what has already occurred. With the understanding that particle-based quantum states can and do combine into linear combinations called superposition states, we can examine how these superposition states combine when particles become entangled with each other. Here's a water wave. It's described by a wave function that determines its operation and a wave equation that determines the change in the function over time. We can channel this wave into two directions, say A and B. With enough time, we can create a great distance between the two branches. If we examine branch A at some time t and find that the wave is at a peak, we will know immediately that at that exact time the branch B wave will also be at a peak. We don't ask how did the A branch inform the B branch that it needed to be at a peak. We did not analyze whether information was flowing from A to B faster than the speed of light. We simply note that both branches are a part of a single wave equation that determines its state at any time t. Of course, if we drop channel A's water off a cliff, the wave in channel B will continue on its merry way. To help isolate the key differences between classical mechanics and quantum mechanics, let's look at one more classical example. Here we start with two coins, each with the heads on one side and the tails on the other. If we put them both into a spin and send them up the two channels, we note that during the journey they exhibit neither heads nor tails, but they carry a probability that, once stopped, they will either come up heads or tails. The probability is 50-50. But unlike the water wave, the results for one of them does not tell us anything about the results of the other. They are independent, but like water waves, the outcomes can be predicted if the starting conditions and channel environment are known. With a quantum mechanics view, we'll start out with two electrons that have been put together to entangle them. Entangled particles are particles that have their quantum states described by a single wave function. The quantum state in question here is the electron's spin. In their lowest energy state, when one is up, the other will be down. Now we send one of the electrons down channel A and the other down channel B. As they travel, they will not exhibit any spin much like the coins did not exhibit heads or tails. In this example, the moment the electron in channel A interacts with a strong magnetic field, it will bring either up spin or down spin to the interaction. At the same instant, the other electron's spin is determined. If A was up, B will be down. If A was down, B will be up. This is as expected because both particles are following the one wave function. It does not matter how far apart the two particles are. In 1935, Einstein, along with Boris Podolski and Nathan Rosen, argued that quantum mechanics was not complete as a theory. He wrote that, to be correct, the theory must match what we observe through experiment and measurement. To be complete, every element of physical reality must have a corresponding element in the theory. Einstein used the following thought experiment to illustrate this point. Consider two identical entangled particles, starting from the same place and moving at the same speed in opposite directions from a common starting point. Letting X represent the distance traveled, X2 would have the opposite sign of X1. Letting P represent particle momentum, and given that the initial momentum was zero, P2 would have the opposite momentum of P1. So their sum would be zero. And each particle has a location and a momentum, means that these quantities are elements of a physical reality. Heisenberg's uncertainty principle rules out the ability to measure these two quantities at the same time for any one particle, because interacting with one of these properties impacts the ability to measure the other. According to Einstein, measuring X2 allows us to predict X1, and measuring P1 allows us to predict P2. With this, we can know both the position and momentum of both particles at the same time. According to Einstein, this is how a complete theory would work. But in quantum mechanics, given that these two particles are under a single wave function, measuring X2 impacts X1 in such a way as to make it impossible to measure P1. From Einstein's point of view, this was spooky action at a distance, and made quantum mechanics incomplete. Einstein proposed that there are hidden variables at play that determine the state of particles like these in advance. One of his examples went like this. Because we have a pair of gloves, one is right-handed and one is left-handed. We place them in two identical boxes and mix up the boxes to the point where we do not know which glove is in which box. Now send these two boxes down to channels A and B. As soon as you open one and find out which handedness it was, you immediately know the other. He thought that someday a new physics theory will uncover these currently hidden variables. Niels Bohr responded with support for quantum mechanics. In his view, reality follows the wave nature of matter without any need for hidden variables. At the time, there was no way to prove whether hidden variables did or could not exist. In fact, how can you even go about proving that a hidden variable doesn't exist? In 1964, an Irish physicist, John Bell, published a mathematical paper proposing a way to test for hidden variables. His work is called Bell's Theorem, or Bell's Inequalities. It was based on entangled electrons and Stern-Gerlich apparatus spin detectors. But we'll use the more easily managed particles, photons, and polarized lenses. Bell's idea was to assume Einstein's hidden variables hypothesis is true and then show how it leads to a contradiction. This would prove that the hidden variables hypothesis is false. The best way to understand Bell's Theorem is to use Venn diagrams from basic set theory. Here's a simple Venn diagram example. Consider the set of all people in a town, say, Paris, Illinois, who go out on a particular rainy day wearing a hat. Some of these people are also wearing gloves. This would be a subset of the whole. Now we count the number of people with hats, and we count the number of people with hats and gloves. If the number of people with hats and gloves is greater than the number of people with hats, you have a contradiction, a violation of the basic assumption. The assumption that they are counting people in the same town on the same day must be false. For example, this violation could happen if the count for hats was indeed taken in Paris, Illinois, but the count for hats and gloves was taken in Paris, France. Bell's thought experiment involves sending photons through polarized filters. If a photon passes through a filter, it is referred to as past. If it's blocked, it is referred to as failed. The probability that a photon will pass or fail depends entirely on the angle between its polarization state and the filter's polarized state. Here we have three tests, A, B, and C. Test A sends vertically polarized photons into a vertically polarized filter. Test B sends vertically polarized photons into a filter polarized at an angle theta. And test C sends vertically polarized photons into a filter polarized at an angle 2 theta. Now the object of the exercise is to examine the role of Einstein's entangled particle hidden variables hypothesis. So we'll use quantum entangled photons along with the assumption that interacting with one of them does not change the state of the other. So all tests start out with vertically polarized entangled photons. The thought experiment used tests in three particular combinations. One was to run a photon through test A, followed by running its entangled photon through test B. The second was to run a photon through test B, followed by running its entangled photon through test C. And the third was to run a photon through test A, followed by running its entangled photon through test C. What Bell was looking for are the number passing test A followed by failing test B called A not B. The number passing test B followed by failing test C called B not C. And the number passing test A followed by failing test C called A not C. Now consider the three sets. Set A of all the tests that passed test A. Set B of all the tests that passed test B. And set C of all the tests that passed test C. Notice where they overlap and where they don't. Here's the subset A not B. And B not C. When we combine them you can see that A not C is a subset. From set theory we know that the number in A not B plus the number in B not C must be greater or equal to the number in A not C. This is the famous Bell inequality. Remember that our assumption is that the states of the entangled particles depend only on their original hidden variables. It cannot change just because there was a measurement taken on the other particle. Being a thought experiment we cannot actually run the tests and count the results, but we can use the quantum state probabilities to compute the results for these three numbers. For an angle of 45 degrees we get 0.75 is greater than or equal to 1. Clearly not true. This is called a Bell violation. It tells us that the assumption that states are determined by hidden variables must be false. The problem is that complex thought experiments like this are filled with assumptions in loopholes. Even in the 1960s there was no known way to build an entangled photon generator. If we could create and manage such photons in large enough numbers we could flood volumes and see entanglement behavior directly. As of now this is not possible. But today we can produce entangled photons at will and see the states of entangled particles change. In 2019 a team of physicists at the University of Glasgow devised an actual experiment that used ghost images to prove quantum entanglement. First we'll cover what a ghost image is and how one is created. Then we'll cover how they proved quantum entanglement via a Bell inequality. Here we have an argon laser sending its output into a beta barium borate crystal. These are unique crystals in that they can turn a photon into two entangled photons. The process is called spontaneous parametric down conversion. A beam splitter separates the photons. One call the idler proceeds through a liquid crystal spatial light modulator. There are many types of such modulators. This one has a thin gold image of the Greek letter lambda embedded in silicon. Given the idler photons' wavelength they will pass through gold and be blocked by silicon. The photons that pass through enter a single photon detector. This detector then sends a signal to the camera. For each photon that travels to the spatial modulator its entangled counterpart called the signal photon is guided to an intensified charge coupled device camera. This is the kind of camera technology we see in modern telescopes. We cover how they work and how far away is it video book chapter on planetary nebula exploding star. There is a delay loop in the photon's path to ensure that it enters the camera at exactly the same time that its entangled counterpart's signal reaches the camera if indeed it did pass through the modulator. The match of one photon with one signal is called a coincidence count. When the camera senses a photon and a signal simultaneously it lights the corresponding image pixel. If the camera gets a photon without a signal it ignores it. As you can see over time the lambda image is constructed. This is called a ghost image. The light that creates it never encountered the object itself. To ghost image a photon's polarization the Glasgow team made some adjustments to this configuration to take advantage of the entangled polarization and the entangled orbital angular momentum created by the beta barium boray crystal. First the image in the spatial modulator is replaced with what they call a phase object that covers the outer edge of a photon's phase plane. This highlights the region of interest. If we ran with just this change we'd see this ghost image. The next step is to introduce a second spatial modulator on the signal path of the photon heading to the camera. If the first angle is zero we get this base image. If we change the angle with a new spatial modulator say one with a 45 degree angle the orientation of the image changes accordingly. This was done for 90 degrees and 135 degrees. Now the key to the experiment is that there is a relationship between the angular momentum of the photon and its orientation that shows itself in the light intensity profile measured as the number of coincidence counts. In other words the intensity features of the ghost image reveal entanglement. The counts show a bell violation proof that there are no hidden variables involved. Therefore we see that the entanglement is real but it is not spooky action at a distance as Einstein proposed. It is just the wave nature of reality as Bohr had proposed. The following quantum eraser experiment was conducted by a team of physicists at the Brazilian Federal University in Minas-Saris. It starts with a normal double slit experiment like we saw earlier but uses counters instead of a fluorescent screen to develop the interference patterns. Here we have a laser that feeds a beta barium borate crystal to create two entangled linearly polarized photons sent off in two directions. In this experiment we call the one direction P and the other S. The photons that go down path P are called P photons and that those that go down S are called S photons. We'll label their linearly polarized quantum states X and Y. Because they are entangled they will travel with probabilities for these states without actually exhibiting them much like the spinning coins, heads and tails. But we know that if the P photon is found to be in state X then we know that the S photon is in state Y and vice versa. The P photons go directly to a single photon detector labeled D sub P. The detector registers the photon and sends a signal to a coincidence counter. The S photons go through a double slit but instead of hitting a fluorescent screen some enter a movable single photon detector D sub S. When it detects a photon it too sends a signal to the coincidence counter. Once the coincidence counter receives this second signal a count is recorded. The counts are tallied for 400 seconds. Then the detector is moved a millimeter and the number of counts in a 400 second interval is recorded for the new detector position. This is repeated until the detector has scanned across a region equivalent to the screen in a normal double slit experiment. The results are displayed by plotting the number of counts as a function of the detector's position. The interference pattern is clearly observed. As we did with the double slit experiment we keep in mind the quantum state of the particles both initial and after the S photon passes through the double slit. Remember that it is the interaction between the two superposition states on the far side of the double slit that creates the interference pattern. Like we did to provide which way information in the double slit experiment we put quarter wave plates in front of each slit. Measuring the polarization at the detector tells us which slit the photon went through. Given the plus or minus 45 degree shifts created by the quarter wave plates the two superposition states cancel each other out. We are left with just two particle like probabilities. When the coincidence counts were tallied at each detector location it was found that indeed the interference pattern was gone. In order to regain an interference pattern we place a polarizer in the P beam closer to the source crystal than the quarter wave plates oriented at plus 45 degrees the same as plate 1 or minus 45 degrees the same as plate 2. This changes the P photon state. The entangled S photon is modified as well but maintains its linear polarity. Therefore it will still be turned into left or right circular polarity by the wave plates and therefore still eliminate the interference pattern generating quantum state terms and therefore still create the blob rather than an interference pattern. But now we will no longer count all the detected S photons. We count only the ones that correspond to P photons that make it through the polarization filter. This will produce an interference like pattern that reflects what is going on with the P photons. This is called a fringe pattern. When we do a run with the filter at minus 45 degrees we get the anti-fringe pattern. Superficially it looks like the situation that prevented interference has been erased. That is why this is called the quantum eraser. But in fact we see that nothing has been erased. When we add these two together we get exactly the blob image we've created ever since we added the which way information. You may have already noted that having the P photon reach the polarizer before the S photon reaches the double slit is irrelevant. The exact same behavior happens if the S photon passes through the double slit before the P photon hits the filter or after. Again fringe and anti-fringe patterns are produced. The setup is made to look like interacting with the P photon changes what happens to the entangled S photon in the past. This has been given its own name, delayed quantum eraser, even though nothing has been erased. Many eraser experiments use beam splitters and adjusted path links to turn the blob into fringe and anti-fringe patterns. Either way I find it very sad that some physicists characterized this experiment as an example of the cause coming after the effect. Developing experiments without loopholes to prove that the entanglement phenomenon is real has always been difficult. But there's nothing like actually using a phenomenon to remove all doubt. Quantum computing is doing just that for quantum linear superposition and entanglement. There is an amazing amount of work around the world going into the development of quantum computers and their subsystems. Here's just three of them. The superposition states and quantum entanglement covered in the preceding segments represent the foundational physics for quantum computing. In order to illustrate how this is the case, we'll actually construct a two-electron quantum computer. The key difference between classical computers and quantum computers starts with their basic unit of information. The classical computers it is the bit, the two values per bit, zero or one. Quantum computers use quantum bits or qubits for short. And because of quantum linear superposition, a qubit has four values. For example, here's a state vector for the spin of an electron. Its position is determined by two angles that define its state. This state can be divided into two base states and two superposition states for a total of four, twice the number of possible values for a classical bit. That's more because of quantum entanglement every time we add a qubit we double the number of classical bits the entangled whole can represent. Here's a table that compares classical computer bits to qubits. Three qubits are equivalent to eight bits, a full byte. The scaling grows into significant numbers as the number of qubits are increased. The real impact comes when we start talking about hundreds or even thousands of qubits. This exceptional scaling for the qubits has a significant impact on the time computer operations will take. For example, let's assume we have a computer with a clock speed of three gigahertz. It could perform three billion operations per second. Let's also assume one operation on one bit for qubit can be done in one clock cycle. This huge scaling potential is what's motivating the development of quantum computers. A bit has to be able to have its settings of zero or one set or changed and have these settings persist over time. Its settings must also be detectable. In classical computers, bits are made of transistors. For a transistor, the absence of a voltage on its control line stops current from passing through making it off or equal to zero. An applied voltage will trigger a current making it on or equal to one. These values are easily set, changed and read and once set they persist for as long as needed. There are a number of ways to create quantum bits, atoms, photons, superconductors, etc. Silicon spin qubits are also promising. A number of companies are working on them. As of early 2022, Intel appears to have the lead with a 26 qubit product. The long term goal is to reach a million. To understand how quantum superposition and entanglement are used, we'll construct a quantum computer out of two electron spin qubits. We start with three layers of silicon. The yellow layer in the middle is made of stretched silicon. It is actually stretched. The distance between the atoms is increased making it easier for electrons to move around. Electrons in this layer will not move up or down into the more compressed silicon without a push. On top we construct an electrically controlled lattice of gates. Negatively biased electrostatic gates in gray and positively biased gates in brown are organized to create two energy wells capable of holding two electrons in place. These two wells are called quantum dots. On top of these two components, we add a micro-magnet to create a tapered magnetic field. This field couples electron spins to the electric field set up by the gates. With this configuration, we can introduce two electrons. The states of these electrons are controlled by microwave and voltage pulses applied to the gates by the quantum computing unit. For example, electron spin can be aligned with a magnetic field in the up or down direction. And the two electrons can also be put into an entangled state by managed exchange interactions across the Coulomb barrier between them. The Hadamard gate is one of the most important operations. It takes a single qubit in a base state as input and outputs a qubit in a superposition state with equal coefficients. The control NOT gate or C NOT gate is heavily used. It takes in two qubits and only flips the second qubit called the target from zero to one or one to zero if the first qubit called the control is a one. Otherwise it leaves the target unchanged. Taking advantage of the fact that the up equals zero state has a slightly lower energy than the down equals one state, a series of microwave pulses will flip the target qubit only when the controlled qubit had enough energy to have measured it as in the one state. And it is done without reading the controlled qubit. Like changing a photon's polarity, this can be done for any number of entangled qubits without disturbing the entanglement state. This is the case for all quantum gates. Measurement is a special type of operation done on qubits at the end of a series of gate operations to get the final values. In a magnetic field, electrons have two discrete energy levels based on their spin. Detecting these energy levels tells us what the spin was. Compared to the gates, measurement is irreversible and hence is not actually a quantum gate. This execution removes the qubit from its entangled superposition state into a zero or one. The results of a measurement are always stored in classical computer bits for analysis. Combinations of quantum gates are called quantum circuits. These combine to execute computer instructions. This is a two-electron spin qubit quantum computer. Quantum dot states are extremely fragile. The slightest vibration or change in temperature can cause them to tumble out of superposition causing errors, lots of errors. That's why, in order to best protect qubits from the outside world, they are housed in supercooled fridges and vacuum chambers. This makes them very expensive compared to classical computers. Because of this, it is expected that quantum computers will only work on those problems that need a gigantic number of bits, jobs like factoring extremely large numbers. Schrodinger pointed out that superposition and entanglement are the two primary characteristics of the quantum world. And whenever particles find themselves close together, they will become entangled, creating unobservable quantum states.