 Welcome to the tenth video in this series on quantum mechanics. Here we'll explore the radical implications for our understanding of reality arising from experimental violations of Bell's inequality. Throughout this series we've seen that the bizarre predictions of quantum mechanics generated considerable debate, most famously between Niels Bohr, who championed the new physics, and Albert Einstein, who, although he played a central role in its development, pushed back vigorously in support of classical determinism and causality. The primary source of quantum weirdness is wave particle duality, as formalized in the wave function description of quantum systems. In 1935, Einstein, Podolski, and Rosen published a paper titled, Can Quantum Mechanical Description of Physical Reality Be Considered Complete? They argue that although quantum mechanics is a very successful description of nature, it cannot represent a final, complete theory. In a complete theory they asserted, there is an element corresponding to each element of reality. They gave the following working definition. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty without disturbing the system. They then presented a thought experiment designed to show that quantum mechanics cannot be a complete description of reality. A simplified version of their argument is as follows. Suppose at the origin of the x-axis, some event emits two identical particles. One travels to the right with momentum p1 in position x1, and the other to the left with momentum p2 in position x2. If the total momentum of the system was initially zero, then to conserve zero momentum, p2 would have to be the opposite of p1, so that their sum is zero. Because the particles are identical, opposite momentum means opposite velocity, so position x2 would at all times be the negative of position x1. Now, quantum mechanics says that the position and momentum of particle 1 cannot both be real, according to the EPR definition, because of the uncertainty principle. If we shrink one uncertainty to zero, either delta x1 or delta p1, then the other uncertainty has to grow to infinity so that their product is never less than Planck's constant. According to quantum mechanics, if the position of a particle is real, then its momentum cannot be and vice versa. Now, suppose we make a measurement on particle 2. If we measure its position, x2, then due to the relationship between x1 and x2, we can predict x1 with certainty without having disturbed particle 1. So, x1 becomes real. If we instead measure the momentum, p2, then we can predict p1 with certainty, again without having disturbed particle 1, so p1 becomes real. In this experiment, reality for particle 1 depends on measurements made on particle 2, and these particles can be arbitrarily far apart. EPR argue that no reasonable definition of reality could be expected to permit this. One has thus led to conclude. The description of reality is given by a wave function is not complete. In previous videos, we've described how in quantum mechanics, measurements are formally represented by applying operators, which we denote with a little hat or a carrot mark. If applying two operators in a different order produces a different result, we say that two operators do not commute, and their commutator is non-zero. Specifically, the commutator of two operators representing quantities that satisfy the uncertainty principle is equal to ih-bar. In the EPR experiment, the commutators of the position and momentum operators for particles 1 and 2 are each equal to ih-bar. Quantum theory says we cannot know with certainty both the position and momentum of a particle. In the language of EPR, the position and momentum of a particle cannot both be real. On the other hand, the operators for the position of one particle and the momentum of the other do commute. Quantum mechanics permits the position of one particle and the momentum of another to both be real. Now form the operator x hat as the difference of the position operators. x hat represents the distance between the particles. Form p hat as the sum of the momenta operators. p hat represents the total momentum of the two particle system. We can algebraically work out the commutator of x hat and p hat, and we find that it's zero. Therefore, quantum mechanics says that for two particles, it's possible to predict with certainty the difference of their coordinates and the sum of their momenta. If we know these values, then measurement of x2 or p2 allows us to predict with certainty x1 or p1 and the EPR argument then follows as before. No doubt about it, the EPR paradox forces us to conclude that either quantum mechanics is incomplete or our concept of reality needs a radical revision. EPR were clearly in favor of the former. A few months later, Niels Bohr published a response under the same title in which he defended quantum mechanics and came down on the radical revision side of the fence. He argued that the wording of the above mentioned criterion of physical reality proposed by Einstein, Podolski, and Rosen contains an ambiguity as regards the meaning of the expression without in any way disturbing a system. By the end of this video, we'll hopefully have a good understanding of this ambiguity. The gist of it is that the necessity of discriminating in each experimental arrangement between those parts of the physical system considered which are to be treated as measuring instruments and those which constitute the objects under investigation may indeed be said to form a principal distinction between classical and quantum mechanical description of physical phenomena. Bohr emphasized the finite and uncontrollable interaction between the objects and the measuring instruments. As we've seen throughout this series, in quantum mechanics we've reached a limit in which the very act of measurement can change the properties of the object being measured. In light of these considerations, Bohr claimed that we are forced into a final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality. This is where things more or less stood for about 30 years. People were primarily concerned with applying quantum mechanics in the laboratory and the real world. This issue, like the Schrodinger's cat paradox, was largely seen as a matter of philosophical interpretation without practical consequences. But it turns out that the debate can be settled experimentally. In 1964, a paper appeared titled, On the Einstein-Poldowski Rosen Paradox, written by John Bell. We're going to follow a simple version of Bell's argument in which we'll, one, define local reality, two, generate falsifiable predictions based on that definition, and three, test the predictions experimentally. What do we mean by local reality, or local causality, or for short, just locality? The ultimate goal of physics is nothing short of a complete description of the totality of existence. Now that's a pretty tall order. In principle, everything in the visible universe interacts gravitationally and electromagnetically. If not, we wouldn't be able to see distant galaxies. But there is no way we could function or make progress scientifically if we continuously had to consider the totality of existence. In practice, we need to be able to say, look, here's a little piece of the universe we'll call a chair. The chair has certain properties we can measure and understand independently of what's going on in, say, the Andromeda galaxy. And here's another little piece of the universe we'll call a frying pan. We can also characterize its properties. If we put the frying pan on the chair, we can describe the interaction of the two objects without having to account for, say, the position of Pluto in its orbit. So we can think of a particle, or more generally an object or a system, as existing in a limited region of space at a given time. We assume that the particle has a state x. This could include things such as mass, charge, momentum, and so on. For a complicated object, it would include a description of all the atoms making up the object and the chemical bonds between those and so on. It's not necessary that we actually know the complete state, only that it exists in principle. All measurable properties of the object are assumed to be uniquely determined by the state x. We measure properties of the particle by performing experiments in a local environment we'll call our laboratory. The principle of locality assumes that in addition to the state of the particle, these experimental results depend only on the local environment in the laboratory during the experiment. Now, a remote event, say a distance DOA, can certainly have an effect on our experiment. But it does so by producing some disturbance which propagates through space at no faster than the speed of light. This disturbance can eventually enter the laboratory where it becomes part of the local environment during the experiment. So it's still true that the experimental results depend only on the local environment. The distant event can have no effect on the experiment in less than d or c seconds. In particular, it cannot instantaneously influence our experiment. Now we're ready to describe a generic experimental setup which under our assumption of locality leads to Bell's inequality. Suppose we take a particle in the state x and subject it to an experiment with two possible outcomes. We'll call this test A and label the outcomes as pass and fail. We have a second test B and a third test C. Under the assumption of locality, the results of these three tests depend only on the local experimental conditions and the state x of the particle. There are eight possible outcomes of these three tests. Using zero and one to represent fail and pass, we can make a table listing all possible outcomes of the three tests and there are only eight possibilities. Under locality, we assume that a given state x corresponds to one and only one row in this table. A particle in the same state x has to produce the same experimental results. If two particles produce results corresponding to different rows of this table, they cannot be in the same state. Now consider a situation where the particle passes test A and fails test B. We can label this as A, B bar, A and not B. That is, A equals one and B equals zero. There are two rows in our table for which A equals one and B equals zero. Now consider a situation where the particle passes test B and fails test C, B and not C. There are two rows where B equals one and C equals zero. Finally, consider passing A and failing C, A and not C. There are two rows where A equals one and C equals zero. Notice that as indicated by these red lines. Whenever A and not C occurs, either A and not B or B and not C must also occur. There are no rows in our table for which A and not C occurs, but one of the other two situations does not. Suppose we took a large number of particles in random states x and subjected them to these three tests. We have counters which keep track of the number of times A and not B, B and not C, or A and not C occur. Then the number of times that A and not B occurs, plus the number of times that B and not C occurs, would have to be greater than or equal to the number of times that A and not C occurs. Because as indicated by the red lines, the last number cannot increment without one of the other two also incrementing. However, it is possible for one of the first two counters to increment without incrementing the third. There are two rows in the table where A and not B, or B and not C occur, but A and not C does not. Therefore we have an inequality. The left side is greater than or equal to the right side. Now the number of times an event occurs in a large number of experiments is simply a measure of that event's probability. So we have that the probability of A and not B, plus the probability of B and not C, is greater than or equal to the probability of A and not C. This is Bell's inequality. Alternately, we can illustrate Bell's inequality using a Venn diagram. Suppose the state x of a particle corresponds to a position on the screen. If that's inside the red circle, then the particle will pass test A, while if it's outside it will fail. Likewise for the test B and test C circles. The region corresponding to A and not B is inside the A circle and outside the B circle, outlined here in white. The B and not C region is outlined in yellow, and the A and not C region is outlined in magenta. The magenta region falls completely within the white and yellow regions, but there are parts of the white and yellow regions outside the magenta region. So the area and probability of the white region plus the area and probability of the yellow region is greater than or equal to the area and probability of the magenta region. However derived, the beauty of Bell's inequality is its simplicity and generality. All we need are a source of particles and three past fail experiments. Bell's inequality is a straightforward logical deduction from the fundamental assumptions of locality. We have assumed that a given particle exists in a definite physical state x, and that the results of an experiment performed on that particle are determined by this state in the local environment of the experiment. If Bell's inequality is violated in any experiment, it must mean that our assumptions of locality are wrong.