The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by the [Einstein-Podolsky-Rosen] correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann’s socks is often cited. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience with Bertlmann, gives the immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is this [Einstein-Podolsky-Rosen] business just the same?
John Bell began his now famous paper with the above paragraph. The Bell Inequality started off like so many other great theories in science – as a simple thought experiment. Its conclusions were not so simple, however, and would lead the way to the end of Einstein’s idea of local hidden variables, and along with it his hopes for a deterministic universe. In this article, we’re going to look at the Bell inequality in great detail. Our guide will be a chapter from Jim Baggots’ The Quantum Story, as it has one of the best descriptions of Bell’s theory I’ve ever read.
Before we start – a quick review:
Niels Bohr’s Quantum Theory says reality is probabilistic in nature. Einstein disagreed, and spent the latter part of his life trying to prove Bohr wrong. It culminated into a thought experiment known as the EPR paradox. This is where Einstein introduced his local hidden variables theory. But John Bell was able to prove that it was possible to show that local hidden variables could not account for all the predictions of quantum theory. Einstein was never able to recover from this defeat and technology would eventually advance enough to test Bell’s theory, proving Einstein wrong. This is the story about how John Bell achieved this remarkable feat.
How To Make A Local Hidden Variable Theory
Let us consider a fragmented pair of hydrogen atoms whose total spin is equal to zero. We send both atoms speeding away in opposite directions. We then measure the spin of one of the atoms using a [Stern-Gerlach] device – which is just two magnets with opposite poles resting very close to each other.
When a spinning particle, or in our case a hydrogen atom, passes between the poles, the direction of the spin can be determined by the direction the atom is deflected. We say an atom with a “spin up” is one that is deflected to the north magnetic pole and designated “+”. Likewise, an atom with an opposite spin will be in a “spin down” state, and will be deflected to the south magnetic pole. Its designation is “-“.
Back to the case of our pair of fragmented hydrogen atoms – when we measure the spin of one we can determine what the spin of the other is, as they have to be opposite. Einstein would say that the spin of each of the atoms was determined at the moment of fragmentation… that there is some type of hidden variable within each of the atoms that determines the spin. Remember that quantum theory says the spin of the two fragmented atoms cannot be known until one is measured. And somehow when one spin is measured, the other atom must know to take the opposite spin. This is the heart of the EPR paradox.
Now, lets look at this local hidden variable idea the way John Bell did. Let us suppose that each atom has a little hidden dial deep within one of its many subatomic particles. A dial that has yet to be discovered. The dial can point in any direction from 0 to 360 degrees. Each of our hydrogen atoms has one of these dials, and when they are paired, the dials point in opposite directions. When the atoms become unpaired, the dials become fixed, as pictured in the figure on the left.
Now let’s pass our atoms through the [Stern-Gerlach] spin detector. With our newfound knowledge of the hidden dial, we find that the atoms get deflected to whatever direction the dial is pointing. Such that if the dial in the atom is anywhere in the top half of the dial face, the atom will have a spin up, or “+” property, and if the dial in the atom is pointing in the bottom half, it will have a spin down, or “-” property. We have created a very basic hidden variable theory. We will soon find out, however, that it’s all we really need to show its limitations compared to quantum theory.
Thinking Outside the Quantum Box
With knowledge of the hidden variables, there is no need for any “spooky action,” as Einstein put it. The dial determines the spin of the atoms at separation. And any measurement of the spin simply determines where the dial was when they were separated. Simple, right? Not quite.
Bell took it a step further and began to calculate probabilities. If the magnetic fields are aligned for both detectors, we can say the probability for the dial to be in the top half of the dial face for Atom A is 50%. Because they have to have opposite spins, we can say the probability of Atom B’s dial to be in the bottom half is also 50%. We denote this as P+- = 50%. As long as the magnetic fields are aligned (as in Figure 20), we can also say P-+ = 50%. This is obvious as the top and bottom halves of the dial face each take up 50% of the total area. But what happens to this value if we rotate the magnets of the Atom B detector relative to Atom A’s detector? When you do this, what is considered the top half of the dial face rotates as well, changing the areas of each dial face half in relation to the dial. Because the dial is fixed into position, it means that the value of P+- will decrease as the angle of rotation increases. It’s easy to visualize if you think of the dial faces overlapping each other.
Consider Atom B at 45 degrees in the figure on the right. Superimpose it onto Atom A. You will see the area that the downward pointing dial can exist in has decreased. The beige colored top face is overlapping the bottom face. Remember that the dial is fixed into position – it cannot change. Do the same for 90 degrees, and you will see the area of the bottom face has decreased by exactly half. As the area decreases, the probability P+- decreases, and the probably of measuring both spins as spin up or “+” increases. With 180 degrees of rotation, probability P+- is at 0%. You now have P++, and we measure both spins as spin up.
A Simple Idea Gives Rise to the Bell Inequality
Quantum Theory gives the value of P+- as 1/2 cos2 (a/2) where a = the angle between the magnet axes. John Bell’s simple hidden variables theory agrees with quantum theory only at angles 0, 90 and 180 degrees, as seen in the graph below. However, considering how crude our theory is, the possibility should exist to make a more elaborate hidden variables experiment that would show that all angles would agree. Is it possible to do this?
After 17 years, John Bell realized that the answer was no, it is not. There is simply no way for any local hidden variables theory to account for all the predictions of quantum mechanics. He would go on to use a character by the name of Dr. Bertlmann, who had an unusual dress sense to eloquently describe his findings. And this will be the subject of part 2 of this post on Bertlmann’s Socks when all will be revealed, at least as much as is possible when it comes to Quantum Mechanics.
The Quantum Story, by Jim Baggott. Chapter 31 ISBN-978-0199566846