May 16, 2021
How to think about quantum entanglement
I have two friends, Alice and Bob. Soon they will be your friends. Since I’m a nice guy, I’m going to give each of them a gift. It’s nothing special — just a box with a couple of buttons and lights on it.
When the left button is pressed, the right light comes on… sometimes. Sometimes the left light comes on. The same is true of the right button. In fact, no matter what button is pressed, the light pattern is completely random.
Alice likes this gift. But she soon wonders — like you I’m sure — how I made this box. What is going on inside that determines the pattern of lights? Maybe it’s a small computer. Or, maybe it just flips coins inside. Perhaps it is a list of instructions.
Bob is wondering the same thing because, as I mentioned, he also has one of these mystery boxes. I gave him a green one, and it also displays a random pattern of lights when the buttons are pressed.
Wait a minute, though! This pattern looks familiar. (If not, scroll up.)
Alice and Bob’s boxes do the exact same thing. And, by the way, it doesn't matter what order they press the buttons in. It also doesn’t matter where Alice and Bob are. They could even be on opposite sides of the world!
Let’s recap. Alice is sitting in a coffee shop in Toronto and decides to play with the cool gift I got her. She thinks about pressing the left button but has no idea what light will come on. She presses it three times and the pattern is right-left-right. She then calls up Bob in Sydney. He also happened to be playing with his box. (I know, I’m a great gift giver.) Oddly, he also freely chose to press the left button three times. Of course, he could not have known what pattern he’d see in advance. But, his pattern of right-left-right is exactly the same one Alice saw. Is this weird?
Correlation and causation
When things happen together, we say they are correlated. We hear about correlations all the time, especially in the science section of the news. We hear things like “sugar is linked to cancer” or “smoking causes cancer”. In either case, what is observed is a group of people that both engage in some activity and have some type of cancer. Obviously, some cancerous lung cells don’t cause a person to smoke, so we default to the opposite. However, establishing a causal link between two things is very difficult. In the case of smoking and lung cancer, it took many years of study to prove this fact. But why? We already said the opposite — that lung cancer causes smoking — is ridiculous. Surely then it is the cigarettes causing cancer, right?
For Alice and Bob, perhaps it is Alice’s box, and the light pattern it produced, that caused the identical pattern on Bob’s box. Maybe Bob’s box has a camera and computer which records and mimics Alice’s box. Perhaps, but remember that Alice is on the other side of the world, and Bob may have pressed his buttons at the exact same instant as Alice. Not even the light from Alice’s box could have had time to travel around the world to reach Bob.
In all cases of correlation, our mind craves to know a causal connection. Like a 4-year-old children, we want to know why. Since we only see the two events, we default to the option that one caused the other. But in most cases of correlation, it is actually an unobserved event that caused both of the correlated events.
A famous example is a fact that a city with more police has more crime. Does more crime lead to more police? Do more police cause more crime? Actually, neither. In fact, a city with a large population has both more police and more crime simply because there are more people, period. This is called a common cause. The reason why very few scientific studies can say one thing causes another is that all possible common causes need to be ruled out, and there are infinitely many potentially hidden common causes. So, to prove that smoking causes cancer was a herculean effort in ruling out all common causes.
Here’s my little secret: I put the same set of instructions in Alice and Bob’s boxes. The instructions were the common cause that led to the correlations of the light patterns.
In the jargon of quantum physics, a common cause is called a hidden variable. Most of the time these are called local hidden variables because the cause has to be traced back to a common location in space and time. Otherwise, we’d break Einstein’s theory of relativity with a cause that travels faster than the speed of light. If correlations persist even if all possible common causes were ruled out, we call them nonlocal correlations.
Spreadsheets
It is easier to list the probabilities for the possible events rather than a potentially unending list of events themselves. There are two possibilities for each of Alice’s button presses, Bob’s button presses, Alice’s lights, and Bob’s lights. That’s a total of 16 possible events. When looking at actual events, some will be repeated. But those happen according to some specification of chance — the probabilities for each event.
Consider a simpler example of a coin. The flips of a coin might produce heads, tails, heads, heads, heads, tails, tails, heads, heads, tails, heads, … But, it is much easier to tell you that the probability of heads is 0.5 and likewise for tails. For Alice and Bob’s boxes, here is an example.
Does this make sense? When they both press the left button, they both see the same light come on, but which light is random. So the light pattern is either left/left or right/right each with 0.5 probability. Whereas left/right and right/left are never seen, so those should each have 0 probability. It all checks out. In fact, Alice and Bob’s boxes never (that is, with probability 0) give opposite outcomes on any button press, not just the left/left button choices. So, the whole story would look like this.
OK, so maybe you don’t like staring at tables of probabilities, but I promise that it’s going to be helpful soon. Remember that any correlation you might encounter is completely summarised by such a table. You’d definitely need to write it down if you wanted to do any further investigation beyond noting the curiosity.
Once we have the table of probabilities we could imagine changing the instructions of the boxes to modify these numbers. Perhaps we might want to make the correlation more subtle, for example. Or, we might want no correlation at all. A completely random pattern of lights that weren’t correlated would have probabilities of 0.25 in every location in the table.
In thinking of the infinitely many ways we might create correlated events this way, we might ask the following very deep and totally profound question.
Can all tables of probabilities be explained by local correlations — that is, by a common cause?
Truly magic boxes
Compare the above correlations with the following very slight modification.
Can you spot it? It’s not so easy to find nonlocal correlations. What would these correlations look like on the boxes?
In words, Alice and Bob’s boxes always light up the same, except when the right button is pressed on both boxes — then the lights are opposite.
This might not seem so special, but here is the important part. Try as you might, with every mathematical tool at your disposal, you will never create these correlations with local hidden variables. That is, boxes that behave this way can’t have instructions programmed inside them.
I see you waiting there for the big aha! moment where I tell you something profound about Nature and why this is so. Sorry, but this fact is simply a mathematical impossibility.
OK, but now you are thinking that I’m going to tell you Quantum Physics allows for such nonlocal correlations. Nope. These correlations are not even possible in quantum physics. So, that’s lesson one: not all correlations can be described by classical or quantum physics.
Quantum correlations
Speaking of quantum physics, here is a table of probabilities that would result if Alice and Bob’s box contained entangled quantum things.
This table also represents nonlocal correlations. If you don’t see why don’t worry — it took quantum physicists over 60 years to notice.
Let’s pause for a moment and recap. Alice and Bob each have a box. Alice takes hers to Toronto and Bob takes his to Sydney. Scratch that. Let’s send Bob all the way to Alpha Centauri, where it would take over 4 years for even communication at the speed of light to happen.
Alice and Bob are asked to press buttons and the outcomes look to be correlated in the way above. Surely they must have conspired before Bob left. They must have rigged their boxes to agree in this way. Alas! Nonlocal correlations forbid it.
The boxes could not have caused each other’s outcomes — they are too far away — nor could there have been a common cause. What’s left? No cause at all, apparently! The outcomes of experiments on quantum things are not caused, but brought into existence.
Reality — some world out there that exists waiting for us to come along and find it out — is an illusion.
Trial by threes
If you are still not convinced, here is another example you can use to prove it yourself. This time Alice and Bob have a new box. Their boxes have 3 buttons and 3 lights.
Whenever Alice presses a button, a random set of lights comes on, but it is always an even number of lights.
Whenever Bob presses a button, a random set of lights comes on, but it is always an odd number.
And, the light on Alice’s box below the button Bob pressed is the same as the light on Bob’s box below the button Alice pressed. So, a sequence of presses might look like this.
Could these boxes have a set of instructions or programs in them to create these correlations? That is, are these correlations local or nonlocal? First, clearly they are correlated since the light on Alice’s box depends on what lights turn on for Bob. But it seems quite complicated. However, we can easily see that the lights "line up” if we tip Bob’s box on its side. The examples above then look like this. The lights are always the same where they overlap.
If we imagine all the ways Alice and Bob’s lights could align, we would form a 3-by-3 square of lights. If inside Alice and Bob’s boxes were squares that gave the pattern of lights to turn on, we would have a local hidden variable explanation.
If a set of instructions existed for these correlations, then it must be given by a list of these squares. Alice’s lights turn on corresponding to the row of the button press and Bob’s lights turn on corresponding to the column of his button press.
So these correlations are local, right? Not so fast. Take a closer look at that square — in particular, the last column. That’s the sequence of lights Bob’s box should display if he had pressed button 3. But, it has an even number of lights on! No worries. Turn that last light off.
Ah! But now Alice’s lights in the last row have an odd number on. In fact, by playing around a bit with this square, you can convince yourself that it is impossible to make any square that works. That is, the only way to program Alice and Bob’s boxes to be correlated this way is to create a 3-by-3 square with an odd number of white dots in each column and an even number in each row. That’s not possible!
This actually proves that the correlations above are nonlocal. And this time, these are exactly the correlations given by quantum entanglement!
Congratulations! You just proved quantum correlations are nonlocal.
