Getting Entangled with Wave Functions

Yesterday, I wrote about research in which scientists teleported quantum information from a laboratory to a satellite in Earth orbit. Today, I would like to describe the basic physics of quantum teleportation and explain a few details about this particular experiment.

And don't worry, it is all far simpler than one might think!

At the heart of this process lies quantum entanglement, and so that is where our investigation begins.

Photo by Spiderplay/iStock / Getty Images
Photo by Spiderplay/iStock / Getty Images

(Thanks to generous contributions, this image is now licensed! Wanna help out, too? Donate!)

Basically, entanglement is just what it sounds like—two or more things are so intimately intertwined that they cannot be taken individually. Instead, it is only possible to consider the whole. To restate this somewhat abstractly, one could say that \(A\) and \(B\) are entangled if \(A\) depends upon \(B\), and \(B\) depends upon \(A\); thus, it is only possible to describe \(AB\) together.

Or, to put it in the simplest terms: Can't have one without the other? Gotta take 'em both!

That's it! That's entanglement in a nutshell. Pretty simple, right? It is straightforward. However, complexity begins to appear as one applies entanglement to quantum mechanics. To fully grasp the concept of quantum entanglement, we must have some knowledge of wave functions, superposition, and probability, and so I must attempt to describe these things to a lay audience. Here we go!

I shall begin by first stating outright that wave functions need not have anything to do with waves. Some of them do, but not all of them. In the example I am going to illustrate below, there is no wave; nothing is waving, nothing is oscillating, the wave function is just a thing, and it does not wave.

So, what does it do? A wave function describes the state of a quantum system, and to illustrate this, I am going to use the simplest example I can think of—that of a common household light switch. In the non-quantum (or classical) realm, this switch can be in one of two positions: either on or off. To reduce the amount of writing we must do when describing our switch's state to someone else, we can use the following shorthand: \[\text{on} = 1,\] \[\text{off} = 0.\] Therefore, if I give you a \(1\), you know my switch is on. If I give you a \(0\), then you know it's off. The switch can be a \(1\) or a \(0\), but it cannot be both at the same time. Makes sense, right? Try it out! Flip (tap or click) the switch:

Now, let us say that we live in the quantum realm. Well, technically, we do! So let us instead say that our light switch is very sensitive to the quantum reality of our universe. In that case, its state must be described using wave functions, the two simplest being: \[\text{on} = \lvert 1 \rangle,\] \[\text{off} = \lvert 0 \rangle.\]

Note: whenever you see a "ket," \(\vert * \rangle\), that is a wave function.

So far, it does not appear that much has changed; we've just put some kets around our ones and zeros. However, this is where the peculiarities of quantum mechanics come to play! Whereas the non-quantum light switch can either be a \(1\) or a \(0\) but not both simultaneously, our quantum switch can have a state which looks like this: \begin{equation}\lvert\psi\rangle\ = \alpha\lvert 1\rangle\ + \beta\lvert 0\rangle,\label{150717:psi}\end{equation} where \begin{equation}\lvert\alpha\rvert^2 + \lvert\beta\rvert^2 = 1.\label{150717:unity}\end{equation}That. What on Earth is that?!

Well, since it's in a ket, we know it's a wave function. Specifically, it is the wave function of our quantum light switch, which I am representing with the lowercase Greek letter "psi." The other Greek letters, "alpha" and "beta," are parameters that tell us how much "on-ness" and "off-ness" our light switch wave function contains.

Wait, what?

That's right, as is often described in terms of cats, *in a way*, our quantum light switch can be both on and off at the same time!

Whelp, we done gone and broke reality.

Whelp, we done gone and broke reality.

How is this possible? Well, technically, it's not. The picture above is simply a joke inspired by Photoshop transparencies, and Schrödinger's cat is either dead or alive, but definitely not both. What equation \eqref{150717:psi} actually says is that the state of our light switch is a (linear) combination, called a superposition, of being on and off. However, when we check our switch to see which position it's in, we will find that it is one or the other—not both. The parameters \(\alpha\) and \(\beta\) represent are the probabilities of finding our switch on or off, respectively. Specifically, \begin{equation*}P(\text{on})=\lvert\alpha\rvert^2,\end{equation*}and \begin{equation*}P(\text{off})=\lvert\beta\rvert^2.\end{equation*}That's why \eqref{150717:unity} is necessary; all the probabilities of a system have to add up to \(100\%\).

Some folks probably find this confusing, especially if it has been a while since they learned about probability in school, and so I am going to try to provide some concrete examples. Say that \(\alpha\) and \(\beta\) both equal \(1/\sqrt{2}\). In that case, \begin{equation*}\lvert\psi\rangle\ = \frac{1}{\sqrt{2}}\lvert 1\rangle\ + \frac{1}{\sqrt{2}}\lvert 0\rangle,\end{equation*}and the probability of finding our switch on or off is \begin{equation*}\left\lvert\frac{1}{\sqrt{2}}\right\rvert^2 = \frac{1}{2} = 50\%.\end{equation*}In other words, if we repeat this experiment a million times, half the time we will find the switch is on, and half the time we will find it's off.

As another example, our switch could have the wave function \begin{equation*}\lvert\psi\rangle\ = \frac{1}{\sqrt{3}}\lvert 1\rangle\ + \sqrt{\frac{2}{3}}\lvert 0\rangle.\end{equation*}In that case, \begin{equation*}P(\text{on})=\left\lvert\frac{1}{\sqrt{3}}\right\rvert^2=\frac{1}{3},\end{equation*}and \begin{equation*}P(\text{off})=\left\lvert\sqrt{\frac{2}{3}}\right\rvert^2=\frac{2}{3}.\end{equation*}Thus, if we repeat the experiment over and over and over again, \(33.3\%\) of the time we will find the switch on, and in \(66.6\%\) of all cases, we will find that it is off.

The point I am trying to make is that the parameters \(\alpha\) and \(\beta\) can take on any value, so long as equation \eqref{150717:unity} remains true. They can be complex numbers, and they can even depend upon other parameters or variables such as time. As long as equation \eqref{150717:unity} holds, then equation \eqref{150717:psi} is an entirely valid wave function that describes the likelihood of finding our quantum light switch on or off.


Is your brain hurting yet? Mine is—but that's because I have been teaching myself how to create and format these pictures and equations on the fly, and that's a pretty steep learning curve. This seems like a good stopping point for our discussion, and so I think I will cut it here for now. But don't worry, we shall pick it back up again, soon! In the meantime, let these things sink into your mind for a while.

Hopefully, I have convinced you that entanglement is easy to understand, and that, at its most fundamental level, so too is quantum mechanics! The math can be challenging, and the interpretation impossible, but the underlying principle is straightforward: quantum systems are described by wave functions which encode the probabilities of different outcomes of the same experiment.

Read that again if you have to: Quantum systems are described by wave functions which encode the probabilities of different outcomes of the same experiment.

This fact is going to be very important next time when we finally get to quantum entanglement, and perhaps even teleportation, too.

If you have any questions, feel free to leave them in the comments. And, as always, stay curious, my friends!