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Dirac Three Polarizers Experiment
In his 1930 textbook The Principles of Quantum Mechanics, Paul Dirac introduced the uniquely quantum concepts of superposition and indeterminacy using polarized photons.
Dirac's examples suggest a very simple and inexpensive experiment to demonstrate the notions of quantum states, the projection or representation of a given state vector in another basis set of vectors, the preparation of quantum systems in states with known properties, and the measurement of various properties.
Measuring a system again after preparing it
Any measuring apparatus is also a state preparation system. We know that after a measurement of a photon which has shown it to be in a state of vertical polarization, for example, a second measurement with the same (vertical polarization detecting) capability will show the photon to be in the same state with probability unity. Quantum mechanics is not always uncertain. There is also no uncertainty if we measure the vertically polarized photon with a horizontal polarization detector. There is zero probability of finding the vertically polarized photon in a horizontally polarized state.
in a known state is a Pauli measurement of the first kind Since any measurement increases the amount of information, there must be a compensating increase in entropy absorbed by or radiated away from the measuring apparatus. This is the LudwigLandauer Principle. The natural basis set of vectors is usually one whose eigenvalues are the observables of our measurement system. In Dirac's bra and ket notation, the orthogonal basis vectors in our example are  v >, the photon in a vertically polarized state, and  h >, the photon in a horizontally polarized state. These two states are eigenstates of our measuring apparatus. The interesting case to consider is a third measuring apparatus that prepares a photon in a diagonally polarized state 45° between  v > and  h >. Dirac tells us this diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes." Thus,
 d > = ( 1/√2)  v > + ( 1/√2)  h > (1)
Note that vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities, as first proposed by Max Born in 1927. When these complex number coefficients are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers represent the probabilities of finding the photon in one or the other state, should a measurement be made. Dirac's bra vector is the complex conjugate of the corresponding ket vector. It is the probability amplitudes that interfere in the twoslit experiment. To get the probabilities of finding a photon, we must square the probability amplitudes. Actually we must calculate the expectation value of some operator that represents an observable. The probability P of finding the photon in state ψ> at location (in configuration space) r is
P(r) = < ψ  r  ψ >
No single experiment can convey all the wonder and nonintuitive character of quantum mechanics. But we believe Dirac's simple examples of polarized photons can teach us a lot. He thought that his simple examples provided a good introduction to the subject and we agree.
The Three Polarizers
We use three squares of polarizing sheet material to illustrate Dirac's explanation of quantum superposition of states and the collapse of a mixture of states to a pure state upon measurement or state preparation.
Here are the three polarizing sheets. They are a neutral gray color because they lose half of the light coming though them. The lost light is absorbed by the polarizer, converted to heat, and this accounts for the (Boltzmann) entropy gain required by our new information (Shannon entropy) about the exact polarization state of the transmitted photons.
In figure 2, polarizers A and B are superimposed to show that the same amount of light comes through two polarizers, as long as the polarizing direction is the same. The first polarizer prepares the photon in a given state of polarization. The second is then certain to find it in the same state. Let's say the direction of light polarization is vertical when the letters are upright.
If one polarizer, say B, is turned 90°, its polarization direction will be horizontal and if it is on top of vertical polarizer A, no light will pass through it, as we see in figure 3.
The Wonder and Mystery of the Oblique Polarizer
As you would expect, any quantum mechanics experiment must contain an element of “Wow, that’s impossible!” or we are not getting to the nonintuitive and unique difference between quantum mechanics and the everyday classical mechanics. So let’s look at the amazing aspect of what Dirac is getting to, and then we will see how quantum mechanics explains it. We turn the third polarizer C so its polarization is along the diagonal. Dirac tells us that the wave function of light passing through this polarizer can be regarded as in a mixed state, a superposition of vertical and horizontal states. As Einstein agreed, the information as to the exact state in which the photon will be found following a measurement does not exist. We can make a measurement that detects vertically polarized photons by holding up the vertical polarizer A in front of the oblique polarizer C. Either a photon comes through A or it does not. Similarly, we can hold up the horizontal polarizer B in front of C. If we see a photon, it is horizontally polarized. From equation (1) we see that the probability of detecting a photon diagonally polarized by C, if our measuring apparatus (polarizer B) is measuring for horizontally polarized photons, is 1/2. Similarly, if we were to measure for vertically polarized photons, we have the same 50% chance of detecting a photon. Going back to polarizers A and B crossed at a 90° angle, we know that no light comes through when we cross the s. If we hold up polarizer C along the 45 degree diagonal and place it in front of (or behind) the cross polarizers, nothing changes. No light is getting through. But here is the amazing, impossible part. If you insert polarizer C between A and B, some light now gets through. Note that C is slipped between A (in the rear) and B (in front).
Let's start by removing the A polarizer from the back. This is the polarizer that prepared the state of the photons in vertical polarization state  v >. The light now comes through polarizer C first, which prepares it in a state of diagonal polarization  d >. We recall from equation (1) that  d > is a superposition of the basis vectors  v > and  h >.
Only some of the diagonally polarized light from C gets through the horizontal polarizer B.
If A crossed with B blocks all light, how can adding another polarization filter add light? It is much less light than through C alone. We shall see why.
The Quantum Physics Explanation
Let’s start with the A polarizer in the back. It prepares the the photons in the vertical polarization state  v >. If we now had just polarizer B, it would measure for horizontal photons. None are coming through A, so no photons get through B. When we interpose C at the oblique angle, it measures for diagonal photons. The vertically polarized photons coming through A can be considered in a superposition of states at a 45 degree angle and a 45 degree angle. Photons at 45 degrees are absorbed by C. Those at +45 degrees pass through C. C makes a measurement of 45 degree photons. It can also be viewed as a preparation of 45 degree photons. Only half the photons come through polarizer C, but they have been prepared in a state of diagonal polarization  d >. The original vertical photons coming through A had no chance of getting through B, but the diagonal photons passing through C (half the original photons) can now be regarded as in a linear superposition of vertical and horizontal photons, and the horizontal photons can now pass through B. Those vertically polarized will get absorbed by B, as usual. Recall from equation (1) that  d > is a superposition of the basis vectors  v > and  h >, with coefficients 1/√2, which when squared give us probabilities 1/2. Fifty percent of these photons emerging from C will pass though B. One quarter or 25% of the original A photons make it through. This happens if we send just one photon through at a time, just as with the twoslit experimant. Just as we can not say that the photon passes through slit A or B (only probabilities are moving in von Neumann’s process 2), we cannot say that our photons are in one state or another. They are in the mysterious linear combination that can collapse instantaneously into one state when a measurement is made.
Dirac's Description
In chapter 1 of his book The Principles of Quantum Mechanics, Paul Dirac describes our experiment. (Complete text of Chapter 1) from section 2, The Polarization of photons, pp.57 Dirac describes the superposition of states with further comments on indeterminacy.
