 Let's illustrate Bell's inequality with a cartoon experiment. Suppose we have a magic brass disc that randomly emits copper bars. Each bar comes out at a random angle and then shoots off through space while maintaining its angular orientation. If we view a bar coming towards us, we can describe its orientation by the angle x it makes with the vertical. We'll treat one of these bars and its angle as a quote, particle in the state x. For our experimental test, we'll employ this slotted brass disc. If a copper bar passes through the disc, then it passes the test. This orientation will form test A, here's test B, and here's test C. Bell's inequality relates the probabilities of passing one test and failing a second. Therefore, we'll use two discs at a time. To keep track, we'll assume one is brass and the other is chrome. We can stack these to create a sequence of tests. Here's test A followed by test B, A followed by C, and B followed by C. A copper rod can pass both tests, fail the first, in which case we'll assume it's absorbed by the disc, or pass the first and fail the second. It's the probabilities of these kinds of events that appear in Bell's inequality. The probability that a rod will pass the first test and fail the second is simply the probability that its angle x corresponds to the visible part of the chrome disc. Here's the range of angles corresponding to pass A and fail B. Here's the range for pass B, fail C, and here's the range for pass A, fail C. We can see that the probability to pass A and fail B, plus the probability to pass B and fail C, is equal to the probability to pass A and fail C. For this cartoon experiment, Bell's inequality is satisfied in its equality form. In the real world, Bell's inequality is often tested using photons and polarization filters, which we introduced in video 9. When polarized light encounters a polarization filter, the emitted intensity depends on the filter orientation. Here's the relationship between transmitted intensity and orientation angle I measured with my inexpensive webcam. Essentially, we'll use this result to test Bell's inequality, although for convenience we'll work with the ideal theoretical form of this curve. A polarization filter operates as follows. Suppose an electric field is oriented at an angle theta to the filter axis. We can think of the field as having one component parallel to the axis, which varies as the cosine of theta, and is transmitted by the filter, and another component perpendicular to the axis, which varies as the sine of theta and is blocked by the filter. The intensity of the transmitted field is the square of the transmitted electric field, hence cosine squared theta. The intensity of the blocked field is likewise sine squared theta. Since a photon has to either completely pass the filter, or be completely blocked, under wave particle duality we have to interpret these expressions as the probability for a photon to pass or fail to make it through the filter. Cosine squared of theta is the theoretical form of this measured curve, and sine squared is simply one minus this. A basic trig identity says cosine squared theta plus sine squared theta equals one. This means that the photon either passes or fails with 100% probability. The average of cosine or sine squared over all angles is one half. The angled brackets here denote an average. This means that randomly polarized photons will pass the polarizer with probability one half. Our three experimental tests are as follows. A filter with its axis oriented vertically forms test A. A filter rotated at an angle theta forms test B. And a filter rotated at an angle 2 theta forms test C. To test Bell's inequality, we combine test A and B and look at the probability that a photon passes test A but fails test B. Assuming the photon is randomly polarized, it will pass test A with probability one half. Having passed test A, it must then be linearly polarized along the axis of filter A. Then the probability that it will be blocked by filter B, which is oriented at an angle theta relative to filter A, is sine squared theta. Therefore, the probability to pass A and fail B is one half sine squared theta. The same process for test B followed by test C gives the probability to pass B and fail C is also one half sine squared theta. Likewise, the probability to pass test A and fail test C is just one half sine squared of 2 theta. Bell's inequality says that the sum of the first two probabilities should be greater than or equal to the third. One half sine squared theta plus one half sine squared theta is just sine squared theta. So Bell's inequality says that sine squared theta is greater than or equal to one half sine squared of 2 theta. A plot of the two quantities shows that this is not true for theta between 0 and 45 degrees. Bell's inequality is violated. If you're uncomfortable with the assumption of a randomly polarized photon, assume we use vertically polarized photons. The probabilities are then sine squared theta, cosine squared theta times sine squared theta, and sine squared 2 theta. For theta equals 45 degrees, Bell's inequality reads three fourths is greater than or equal to one, which is false. Now this is a profound result, so let's exercise some skepticism and really think about the details of how we would perform this experiment. We're going to assume ideal filters and detectors. In the real world, things are imperfect, and typically additional steps need to be added to experiments and analysis to compensate for this. Instead of simple polarization filters, we would want to use polarizing beam splitters. We discussed these in video 9. If a beam splitter performs test A and the photon passes, then it continues on in a straight line. If it fails, instead of being absorbed, it's deflected and falls on a detector. If the photon passes test A, then it encounters a beam splitter for test B. If it passes, it continues in a straight line and strikes a second detector. If it fails test B, it's deflected and falls on a third detector. For every photon entering this system, one detector registers a signal. The fraction of those signals produced by the middle detector is our measure of the probability of passing test A and failing test B. A similar setup can implement test A followed by test C, or test B followed by test C. Doing this will show the violation of Bell's inequality we've already discussed. However, there's a loophole. In deriving Bell's inequality, we assume that the tests are performed on one or more particles in the identical state X. If a photon enters our device in the state X and passes the first test, can we assume that it's still in the state X when it encounters the second test? Let's remember Bohr's warning about the finite and uncontrollable interaction between the objects and the measuring instruments. Maybe a photon in state X that passes through a polarization filter comes out in a different state Y. If so, then we can't use that photon for additional measurements without violating the assumptions of Bell's inequality. In fact, we can show that this must be the case. The light emitted by my LCD monitor is linearly polarized. Here I've oriented a polarization filter to block 100% of this light. If we insert a second polarization filter, however, we can actually get some light to pass through both filters. Assume the photons emitted by the screen are in state X. As shown on the left, 100% of such photons fail to pass filter D. As shown on the right, if they first encounter filter C, some pass and subsequently go on to pass filter D. But if they eventually pass filter D, they cannot be in the original state X, because all photons in state X are blocked by filter D. They must come out of filter C in a state other than X. So when we measure the polarization state of a photon, we cannot assume it ends up in the same state that it started in. Well, did we just waste several minutes? No, we're going to fix the loophole, and this will all tie together. But first let's go back to our cartoon experiment. We might imagine that as a copper bar is passing through a brass disc, it experiences some interaction that changes its state, its orientation. Consequently, performing two tests on the same bar may not give us the same results we would have obtained without the interaction, or that we would obtain from independent tests on identical bars. But suppose our copper rod source produces two rods traveling in opposite directions at the same angle X. Now if a measurement on one rod changes its state, we have a copy for the second test. We'll run our two tests on the two identical rods. Because no rod is used for more than one test, we no longer have to worry about changes of state due to a measurement interaction. A real-world version of this takes the form of so-called entangled photons. One of the first proposals for generating entangled photons was through the decay of positronium. Positronium is similar to a hydrogen atom but with the proton replaced by a positron, the electron's antiparticle. In the lowest energy state, positronium, like hydrogen, has no orbital angular momentum. If in addition the positron and electron spins are oppositely oriented and the atom is at rest, there's no net linear or angular momentum in the system. Eventually the antiparticles will annihilate each other and radiate two photons. In order for the photons to have no net linear or angular momentum, they will have to have the same wavelength, travel in opposite directions, and the angular momentum due to the combined spins will have to be zero. If for example one travels to the left with a right-hand polarization, the other will travel to the right with right-hand polarization. Their angular momentum vectors point in opposite directions and cancel out. This creates a scenario very analogous to the Einstein-Pedalski and Rosen thought experiment. Actual experiments employ more complicated but more practical methods for generating entangled photons but the basic idea is similar. The experimental evidence that the two photons are entangled comes from subjecting both to identically oriented polarization measurements. If one and zero represent pass and fail, then each photon generates a pattern of zeros and ones. The two patterns, although random, are identical. If the two photons were generated in different states, we would not observe this. At least some of the time the patterns would be different. In fact, this is so reliable that it's used in the field of quantum cryptography to generate a shared random key for two users. The beauty is that for a third party to intercept the key, they'd have to detect one of the photons, which destroys it, and that'll then be noticed by the intended users. To test Bell's inequality, we subject the two photons to different polarization tests, such as A and B that we've shown here. By counting the fraction of the time that the A and not B detectors generate coincidence signals, we can measure the probability of A and not B. This has been done with a large separation between the photon measurements. Experimental results consistently show that Bell's inequality is violated, just as in the single photon scenario. We could explain this away when using a single photon to perform two tests. The first measurement changed the photon's state, so the second measurement didn't conform to the assumptions of Bell's inequality. But if we want to use that explanation for the entangled photon version, then we have to assume that a measurement on one photon instantly changes the state of the other photon. The change has to be instant, because the two measurements can be arranged to be arbitrarily close in time. Indeed, if they're simultaneous, then we'd have to ask which measurement was influencing the other. This is precisely the picture that the EPR paradox found unacceptable, where reality for one particle depends instantaneously on a distant event, what Einstein referred to as spooky action at a distance. In our cartoon experiment, we might visualize this as a measurement on one rod, causing both rods to simultaneously change their state. Alternately, maybe thinking of each photon as independently in a state X is invalid. Reality in this sense is non-local. Indeed, this gets to the heart of Bohr's response to the EPR paradox. EPR stated, a sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty without disturbing the system. Implicit in the thought experiment was the idea that one can measure the position or momentum of particle 2 without disturbing the system. The system in this view is particle 1. But quantum mechanics requires that the system include both particles, which together are described by a single wave function. It is not possible to make a measurement on either particle without disturbing the entire system. This is the ambiguity in the EPR definition of reality, which Bohr pointed out. As stated by Alan Aspay, the experimental violation of Bell's inequalities confirms that a pair of entangled photons separated by hundreds of meters must be considered a single non-separable object. It is impossible to assign local physical reality to each photon. The experimentally confirmed predictions of quantum mechanics are very clear and fundamentally incompatible with our common conception of locality. Somewhat frustratingly, given our innate desire to understand the world around us, the theory doesn't replace this with a readily understood picture of reality. Consequently, while the theory provides highly accurate and unambiguous predictions for a wide range of physical phenomena, there is no widespread consensus on how to interpret the theory. Following Richard Feynman it seems safe to say that nobody understands quantum mechanics.