So, if you don't know anything about the quantum state, then measurement gives us very little information about it. But if you do know something, then we can attune our measurement process to obtain more. Let's see an example. Imagine that we know from a trusted source that our qubit state is either this vector or this vector. This will go phi, phi is the sum of zero and one. Remember, here we have vector zero and the vertical vector is vector one on this graph. This we got t and t is one divided by square root of two, one minus zero. So, we know that our quantum state is either this or this. Then, measurement of this qubit in the standard basis will give us zero additional information because for both these states, the probability of outcome one is the probability of outcome zero they're equal and it is one half. So, if you obtain one as measurement outcome, it gives us nothing because it has the same probability for both vector psi and phi and the same is for zero. But we can choose another basis to obtain more information from this state. I'm going to choose the basis, we shall draw in green here. So, these basis has also two vectors of course and the first vector we'll call plus, this vector and this actually the same as vector phi and vector minus which is here is this. This basis is called the Hadamard Basis in the name of French mathematician Jacques Hadamard. Now, these two vectors are orthogonal, they are unitary, so they form the orthonormal basis of course. If you measure all the system in this basis, then result plus will tell us that the system was and still is in the state phi and minus outcome tells us that the system was instead psi, before the measurement. So, choosing the correct basis is crucially important for obtaining information from quantum system. Though we don't have the state of the art quantum hardware here, we can still illustrate the measurement process. Remember I told you about electromagnetic wave that we can carry qubits coded by the polarization of photons. The polarization is the direction of the wave oscillation which is orthogonal to the direction of wave propagation. Fortunately, we have crystals which can detect and alter this polarization, this characteristic of a photon and on molecular level, these crystals are organized like this. They contain dipoles which are all oriented along one axis. You see these little springs here on the picture, actually this picture is not exactly correct, there are no springs, there on the molecular level but the idea of spring there exist. If we apply an electromagnetic field alternating electromagnetic field with the vector of oscillation parallel to this axis, then these charged particles pluses and minuses begin oscillate around the positions of the minimus of potential energy. As you know, when charged particles oscillate they produce waves, so this another wave, secondary wave is depicted here, in blue. It's not hard to show that this wave, this secondary wave will be shifted half of the period to the primary wave if this primary wave is for example sinusoidal. Here on this slide, you can see the picture of the fallen wave here, we see this crystal from the side. So, the following wave falls into the crystal, it passes through, it is depicted in orange. Here we can see the secondary wave in blue and of course it propagates in both directions. This we shall draw in green, it is a reflected wave and this wave which is orange is primary and blue is secondary, they're shifted half of the period so they adopt resulting and no wave here. So, the wave doesn't pass the crystal if it is oriented along this axis. On the contrary, if you have the wave polarized like this, then it does not cause oscillations of this charged particles so it doesn't produce the secondary wave and the wave easily passes the crystal. So this is why some materials are not transparent for some wavelength. If inside of the material there is some charged particles which can oscillate with a frequency of this falling wave, then these charged particles produce secondary waves and they self-destroy the secondary wave, self-destroy with the primary wave. So wave does not pass through this material and this fully reflected. This is why all metals are not transparent for almost all frequencies of the electromagnetic waves because of free electrons there and also this is why they are so shiny because of these reflected waves here. This is probably another surprise to you. So if you have this axis which shows the waves to pass, it's called the optical axis of the crystal. So, I draw here the optical axis, this is the axis for which all waves polarized along this axis. They are allowed to pass through the crystal. This axis is orthogonal and all waves which are oriented like this polarized like this, they don't pass the crystal. It is actually the line along which all the dipoles are oriented. If you have a wave polarized like this with some angle theta with the optical axis, then it is not probably a surprise to you that the wave passes through the crystal with probability cosine square Theta and it doesn't pass with probability sine square Theta. So these crystals can perform for us a measurement of qubits which are encoded by the photons polarization. These crystals, linear polarizers are easy to get. They are used in many modern devices. For example, the surfaces of almost all LCD screens have several polarizing films on them, but if you want to get something this clear and transparent, then you probably have to go to a shop where photographers buy their stuff. The reason why photographer use these things, these linear polarizers is this: Many surfaces when they reflect, the light also polarize it. So you can either enhance this polarized light with linear polarizer or to reduce it. For example, if you want to take a picture of a blue blue sky, so we want to enhance the blue color of the sky, you're going to use this thing to enhance this blue color because the light reflected from different layers of atmosphere is polarized. If you want to take a picture of a shallow pond or the bottom of a shallow pond, then you probably would have a problem with the rays of light which are reflected from the surface of the water because they mix with the light which is reflected from the bottom of the pond and they spoil the image. So with this polarizer, you can also restrict this reflection from the surface of the water because it slightly polarizes it. If you have two of these things, then we can do even more because you understand that the light which pass through the polarizer is polarized along its optical axis. So if you cross the optical axis of polarizers, then the light which came from the first polarizer is restricted by the second polarizer like this. So now I hope you don't see me. Now you do. Aside of these useful application of hiding me from students, there are many more. Almost all transparent materials alter the light polarization. So if you put something transparent between these two crossed polarizers, then the light which came through this transparent material can now pass the second restriction polarizer because its polarization slightly changed. If you have some higher densities in this transparent material, then the light there is shifted, the light polarization is shifted more. So on these areas where we have higher densities, we will have more light which is not restricted by the second polarizer. Here on this slide, you can see the picture of ordinary glasses which is put between two polarizers. So, it is a picture in the polarized light and you can see some tensions here for example on the place where glass is attached to the glass screen, there are more light because there are some tensions which the glass has in this area, and thus we can analyze transparent materials. We can see the defects in them which we don't see in normal light. So, we can now understand that this polarizers can perform the measurement process for us for the light polarization and the choice of the basis is performed by rotation of the polarizer. So, the choice of the basis is rotation of polarizer and the light which pass through the polarizer is now polarized along its optical axis and the light which is fully reflected is polarized perpendicularly through its optical axis.
