During a lecture for Introduction to Quantum Computing, the topic of the Quantum Zeno effect came up. We’ll get into the details of this, but essentially it says that — unlike a “watched pot” which eventually boils — a “watched quantum pot” never does boil. One of the students asked if this was similar to how Boo works in Super Mario Bros. The answer is almost, but no. However, we can use Boo to learn about quantum physics anyway!
Only 90’s kids
Nineties kids and younger millennial hipsters will have fond memories of Boo, the lovable ghost character introduced in Super Mario Bros. 3, the greatest video game of all time. If you are not familiar with Boo, you can see the game mechanics in this video:
Here’s the gist. If Boo touches Mario, Mario loses his power-up or dies (don’t worry, he comes back to life). Boo chases Mario if Mario is not facing Boo. If Mario is looking at Boo, Boo covers its face and doesn’t move. I made a simple animation summarising this.
Characters in video games obey rules. These rules are coded by the developers intentionally — if the game behaves in a way not intended, it is called a bug. If you looked at all the lines of code needed to make a video game, you would be able to see the rules. Sometimes these rules are called physics. In some games the “physics” is actually real physics. For example, in Angry Birds, the rules the birds obey are the actual equations of motions invented by Isaac Newton over 300 years ago to describe gravity. The “physics” of the game is the physics of real life. Mario, on the other hand, doesn’t obey the true laws of gravity when you make him jump. But, there was still a rule the character obeyed that we could call “physics”. In any case, all of the rules coded into nearly every video game ever made are derivatives of Newtonian physics. Newton’s laws are often called classical physics — the reason being that it has been superseded by modern physics, including quantum physics.
What if the rules Boo obeyed were those of quantum physics?
Quantum Mario Bros.
Disclaimer: if Boo obeyed quantum physics, we would not be able to see it do much of anything interesting on a screen because a screen displays classical information. The purpose here is to give you an intuition for some aspect of quantum physics. For a deeper understanding of quantum physics, you have to read this book.
With that in mind, we ask: what if instead of super Mario, we had quantum Mario? Actually, scratch that — let’s keep Mario classical. He will act as the scientist — or, as they are called in the quantum business, the observer. Instead, let’s make Boo behave quantumly.
The first difference between Classical Boo and Quantum Boo is that Quantum Boo doesn’t move along a smooth path from start to finish. Quantum Boo is never found in the space between the starting point and Mario. Quantum Boo can only be found at its starting point or on Mario. The only thing that changes for Quantum Boo is the probability of catching Mario. This probability can be visualised as transparency. The more transparent Boo is, the less likely it is to find it there.
Quantum Boo starts to disappear from its starting location and simultaneously reappears at Mario’s location. After some time, Boo’s probability of catching Mario is 100%. Before then, there is some chance that Boo has caught Mario and some chance that Boo hasn’t moved at all. At this midpoint, Quantum Boo is in a superposition of both locations.
Uh oh! Quantum Boo seems unstoppable. But wait! Mario has the power of observation — the power of quantum measurement. Whenever Mario looks toward Quantum Boo, Boo is only either found on top of Mario or at Boo’s starting position. If timed correctly, Mario can force Quantum Boo back to its starting position with a 0% chance of catching him. Of course, Mario could get unlucky and turn to find Boo on top of him as well — it all comes down to a quantum coin toss!
When Mario is not looking, Quantum Boo begins the transition. But this takes time. And, if Mario forces Quantum Boo to the starting position, the clock resets and Quantum Boo has to start the transition all over again. So, if Mario turns around often enough, he can force Quantum Boo to never move at all!
Well, that’s it. That is how a quantum Mario Bros. game would work. But now that your appetite is whetted, I know you want to dive deeper. So what are these principles of quantum physics on display in quantum Mario Bros.?
A ghostly state
In classical physics, the state of Boo is a list of all the important properties it possesses. Boo has a location, direction, speed, and whether or not it is hiding. Boo’s position can be anywhere, which means the possible states are continuous. Continuous means any small change is another allowed state. You can’t count the number of possible states in classical physics — it’s uncountably infinite. In quantum physics, the possible states Quantum Boo can be found in are discrete and finite — you can count them.
This is generally true in quantum physics, and the first identified departure from classical physics. In fact, a Nobel prize was awarded for this “quantum hypothesis” made in 1900. Now, if Quantum Boo can only be found in one of two locations, it must jump from one location to another without visiting the space between. Many of the early physicists studying the new quantum theory famously despised these “quantum jumps”.
Super Position Bros.
Yet, Quantum Boo does somehow move continuously between the two locations in an ephemeral way. The probability of being in one location or the other changes continuously. An instant after Mario turns his head away from Quantum Boo, Boo ceases to be in the starting position. Boo is also not at Mario’s position either at this point. In fact, Boo is in no definite position at all! This new state is called superposition. It’s not here or there, and it’s not both here and there. It is something entirely new.
A lot of words are written about quantum superposition that are just plain wrong. Here they might have said Quantum Boo is in two places at once. But this is wrong — Quantum Boo is at neither location, so it can’t be at both! When words are written that are correct about quantum superposition they are often contorted in unnecessary ways, as if they are skirting around some issue. The reason for this is the apparent need to always couch quantum physics in classical terminology. Well, here is a better analogy for quantum superposition. Imagine you have blue and yellow flowers. You’ve always had blue and yellow flowers and that is all you know. One day, you decide to cross-breed them. You end up with a new color of flower. What do you call it? Do you call it blue and yellow? No. Do you waste an entire blog post about explaining why it is neither blue nor yellow? No. You simply give it a new name, green. That’s quantum superposition. You can’t contort it into classical physics language — it’s just something new.
Phew. I didn’t mean to go off on that tangent. Now, where were we?
Quantum Boo gets a classical scare
Mario cannot “see” the quantum superposition state of Boo. Boo only occupies a superposition when Mario is not looking. There is no answer to what quantum superposition “looks like”. As soon as Mario looks, Boo is either found on top of him or way back at Boo’s starting location. Boo is never found in the space between. In quantum physics, Boo is said to have collapsed into the state it is found in.
This is the source of the so-called measurement problem in quantum physics and at the heart of all the meta-physics and philosophy under the name quantum foundations. In short, the problem is that there are two rules for how Quantum Boo behaves. When no one is looking, Boo seems to spread out potentially occupying every allowed state. But, when someone decides to look, Boo jumps instantly to one of those states. Some physicists say that this is a problem because laws of physics should apply independent of whether physicists, or Italian plumbers, decide to look. After more than 100 years there is still no consensus on this problem beyond the fact that quantum physics works impressively well when applied in the laboratory.
Intermission: who the hell is Zeno?
Zeno was the dictator of the galactic confederacy who brought billions of Teegeeack to Earth in a souped-up jumbo jet 75 million years ago only to kill them with hydrogen bombs releasing thetans which now stick to humans and cause spiritual harm. Wait. No. Wrong book. That was Xenu, not Zeno. My bad.
Zeno (of Elea) was an ancient Greek philosopher known especially for his “paradoxes”. Most of Zeno’s paradoxes are little arguments for the impossibility of motion. Of course, we move all the time — hence the paradox. The most relevant one is the “Arrow paradox”, which can be simplified as follows. At every instant an arrow is motionless (it is where it is). It takes time to move. But time is composed of instants. So it is always motionless. Therefore the arrow does not move.
There have been many arguments given about this paradox that I won’t repeat here. The simplest refutation is to deny that time is composed of instants of zero duration. Now back to our regularly scheduled programming.
Would you believe me if I told you that now you have all the knowledge and intuition to explain the quantum Zeno effect to your friends? (Maybe don’t though unless quantum physics naturally comes up in conversation.)
Boo starts at 100% probability of being in the starting state and that number smoothly goes to 0%. After a few instants then, the probability of being in the starting state is still, say, 99%. If Mario turns around, there is a pretty good chance that Boo is found in the starting state. When Mario turns away, Boo moves again into slight superposition, slowing increasing the probability of landing on Mario. But, if Mario turns around again quickly, Boo will surely be found back in the starting state.
In other words, if Mario turns frequently enough, he can ensure that Quantum Boo never moves. It’s the quantum Zeno paradox, as taught you by quantum Mario.