Guess Who: an Intuitive Approach to Quantum Mechanics and its Spooky Math

Guess Who: an Intuitive Approach to Quantum Mechanics and its Spooky Math

Quantum Mechanics is a buzzword that gets thrown around a lot, but whether you’re a grade twelve student who has to go through SCH 4U or simply someone who’s curious, it’s important to get a good understanding of the basic principles. So without further ado, I present Quantum Guess Who: the most convoluted and potentially rewarding analogy you’ll ever get regarding quantum mechanics.

The Beginning

So quantum really starts off with this revolutionary idea by Max Planck:

What if light was just tiny bits?

Max Planck, maybe.

In any case, the idea of light being ‘quantized’ or split into discreet particles with constant values called photons is what started this wacky field of science. Something strange happened when scientists started looking at light this way…when we tried to see if it was a particle, light stopped behaving like particles! If you’re interested, there’s an entire long history of dispute with Newton, Einstein, and many of physics’ bests which you can read here.

It turns out that when things get really, really small, it gets really, really hard for us to intuitively understand those things. If you were asked on a test what sort of work you exert on a textbook when you lift it, it’s quite easy to imagine. If you were asked on a test where the electrons are around the nucleus of an atom, you would probably not be able to pass that test without studying. In fact, questions like these: where, when, how fast, which direction, etc… may all seem like trivial and easy to answer questions in the classical world but in the quantum world, the answers to all these questions are uncertain.

Superposition

A cat peaking out of a box. An illustration of Schrodinger's Cat.
An artistic representation of Schrödinger’s cat: the popular thought experiment used to demonstrate superposition. You may notice that with the cat head peaking out, it sort of destroys the idea of superposition… I took some artistic liberties.

Because quantum wackiness exhibits strange qualities of both wave and particle, it is impossible to predict accurately the information within a particle. Often we get a distribution of probabilities, readily seen in Guess Who such as you have three characters on the first row, five characters on the second, two on the third, and one on the forth. Now we say the particle (or character) is in a superposition of these different states. When you make a measurement or ask the opponent a question such as “is my character Bob?” you eliminate all other characters or possibilities and the game has ‘collapsed’ onto the current state of Bob. If you have ruled out a character incorrectly or asked the wrong questions, that information is lost to the game. This principle is essential to understanding a quantized system where when a measurement is made, the entire system collapses onto a certain state, information is lost and if you measure it again, you’ll get the same value you got the first time.

The Qubit

To help you illustrate this concept of superposition, let’s talk math. If you don’t know already, computers run on a binary system, where all the information stored on it is represented in 0s and 1s. If we extend that idea to the quantum bit or qubit, we can imagine a quantum particle that’s in a superposition of two states. Drawing on our Guess Who analogy, this is akin to having a board of Guess Who characters and arbitrarily categorizing the characters as the ones with hair and the ones without. We represent this in the following vectors:

a qubit of the basis 0.
zero, binary off, or no hair

a qubit of the basis 1.
one, binary on, or hair

In fact, you may have noticed that with some manipulation, you can represent any vector on the Cartesian plane with some combination of these two vectors. (6, 2) = 6 |0> + 2 |1>, and so on and so forth. To make this more intuitive, all the Guess Who characters on the board can be represented by some combination of the probability of people with hair and people without. If you let the bald characters be represented by |0> and with-hair characters be represented with |1>, you now have a way to describe your game with linear algebra!

The |1> and |0> are called an orthonormal basis:

Ortho from orthogonal, or perpendicular. This means that no aspect of |1> can be represented with |0> and no aspect of baldness can be represented by non-baldness.

Normal meaning normalized, such that all the probabilities of |1> and |0> in a basis where each term is multiplied by (1/sprt2) add up to 1, just as the probability of with-hair and hairless characters add up to one in a game of Guess Who.

Finally basis, meaning that with these measures, all the combinations within a given set can be satisfied using only this ‘basis’. All the characters can be sorted into some distribution of bald and not bald, just as all the 2D vectors can be represented in some combination of the two states.

Orthogonal bases are key in representing qubits as they are just probability distributions of either two bases. In fact, there are many bases and the three common bases:

|1> and |0> |+> and |-> |+i> and |-i>

make up something called the Bloch Sphere where all the probabilities of any possible state of a given two-dimensional quantum information system can be modelled geometrically. I won’t tell you what the other bases are defined as but it turns out you can represent the bases in terms of each other – a fun googling activity for you to do after reading this article!

Matrices

So in quantum experiments, you might often see the use of mirrors, beam splitters, phase shifters, and different things that might affect the state of the qubit or quantum property. Note that this is not a measurement of the state, but it is mathematically equivalent to applying some matrix transformation on the vector. To address matrix multiplication, a little understanding of linear algebra is required. For the purposes of this article, just understand that matrix multiplication is like asking a question in Guess Who. When you ask a question, you transform the game field and the way in which the probability is expressed. Sometimes, the questions you ask may do nothing at all to the number of characters on the board. In that case, the characters you have asked the question to are the eigenvectors, and they will not change through a given matrix transform (save by a scalar factor).

Composite Systems

So what if you have a system of more than one property. What if you would like to know information on both baldness and gender or both the x-plane and y-plane. Well, it turns out that you can also represent this using something called a tensor product.

So say I have |1> and |0>, the tensor product of those two would be |10>, and it is represented as a four-dimensional vector. Similarly, the information regarding baldness and gender can be represented together as a tensor product of the two, or it may be factored out and represented as its two separate bases: bald or not bald, and female or male.

But what if there exists an example where one piece of information about my character cannot be separated from the other?

Entanglement

Entanglement happens when you have two pieces of information to analyze, such as the hair colour of your guess who character and whether or not they have hair. These two pieces of information have some sort of correlation, in other words, if you are able to determine some information about one thing, then more information is revealed about the other. This is the case when you ask if the character has hair. If your opponent says no, then you know immediately your character does not have black hair.

In an entangled state, no matter what the opponent answers, you learn something about the other qubit from your initial qubit. This sounds confusing but mathematically, a pair of entangled particles exist in a state which cannot be factored into the tensor product of two pure states. In other words, the existence of either state depends on each other.

This is where our Guess Who analogy starts to diverge from our quantum strangeness. In any case, if you can suspend your disbelief for a minute and imagine that we are playing Guess Who with qubits rather than cartoon characters, then you’ll probably just see that this ‘entanglement’ business truly transcends any form of intuition.

Entanglement can occur at a distance and exhibit different levels of correlation and anti-correlation. If you would like a nicer “Alice and Bob” introduction to quantum entanglement, Philip Ball does a pretty good job here. For the purposes of our analogy, technically when you ask any question at all, due to the nature of Guess Who, you will eliminate certain possibilities and probabilities by asking a random question. In fact, the game is predicated on the ability to gain the most information from the least amount of yes and no questions so the concept of entanglement is really intrinsic to the system! How cool.

Uncertainty

So this is the part where we admit we don’t know anything. Actually, that would be a lie too… we know some things. And the SUM of those things is a finite quantifiable value dictated by Heisenberg’s uncertainty principle. Since light is somewhat like a wave it exhibits the property of wave-like systems where all the knowable physical properties of the system are limited to a certain amount. This means that if you know more about one aspect of the system (say the speed of a quantum particle), the less you know about another (maybe the position).

In Guess Who, this is readily seen in the number of turns you have. This value is finite, and although it does depend on how efficiently you use it, the more you learn about your character’s face shape, the less number of turns you have on determining their nose shape. This means that the more certain you are about one aspect, the less information you can gain about another. Just like a qubit!

Why do I Care?

So this article has been pretty theoretical so far, let’s bring it back to the application side of things. With a basic understanding of the uncertainty principle, can you now think of uses for this technology?

A great effort has been put into using quantum mechanics to secure information. If a measurement tampers with information, then a clear logical line can be drawn to use it to prevent and detect eavesdroppers as their measurements will collapse the data randomly. This means that the hacker will not be able to access your data and you will know when someone has interfered with your transmission. The field of quantum cryptography is pretty cool and you can learn more about it here.

Finally, the world is your qubit: uncertain, weird, and sometimes freaky, but full of potential! The strange nature of quantum mechanics can lend itself to time travel theories (more on that in our upcoming podcast), understanding how life began in this universe, detecting early cancer signs, proving the existence of gravitational waves, and quite literally anything!

So now that you sort of understand the hazy intuition behind wacky small things, go forth and dominate the world of tiny! Make those small things solve big problems.

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