Anyone can understand quantum mechanics — Part 2

In Part 1 of this series I made the bold claim that, unlike what famous figures in science seemed to suggest, quantum mechanics is a beautiful and simple theory that is accessible to anyone who is enthusiastic about learning it. In Part 2, I am going to put my money where my mouth is and teach you the basics of quantum mechanics in four short lessons. Sounds good?

Before we begin, there are two important points I need to clarify.

  1. I need your help! It would be great if we could learn new things passively and without any effort, like Neo in the first Matrix movie.

kung fu

Unfortunately, the technology for instant learning is not currently available, so we’ll have to rely on old-school methods like reading, thinking, and discussing. For best results, I recommend that you find time during a quiet afternoon, make yourself a coffee or tea, sit down in a comfy chair and go through the material with patience. If you do it together with a friend or loved one, even better. The ideas I will present are not really complicated, but they will be unfamiliar and unlikely to sink in if you don’t help me with your full attention and enthusiasm.

  1. There will be math! I know that the fear of mathematics is a widespread malady that would normally make many of you run away from this post as quickly as possible. My wife, for example, automatically starts yawning when she hears the word “probability”.
Math fear.jpg

There’s math in the post? Noooooo!

I can assure you, there is nothing to fear. Many science writers choose to get completely rid of math when discussing quantum mechanics, probably hoping to increase the sales of their books. This leads to a tragic state of affairs where any explanation of the subject is either full of technical jargon – as in standard university textbooks – or it is carried out with imprecise analogies and hand-wavy arguments. This would be understandable if the mathematics were actually hard – like in general relativity – but in quantum mechanics, we don’t really need to do much more than add and multiply numbers. I won’t insult the intelligence of my students by pretending they will run away just because we’ll use the same type of math they already use when they need to figure out how much money to take on their next vacation. Just stick with me and you’ll be fine.

Lesson 1: What is quantum mechanics?

Surprisingly, quantum mechanics is not a physical theory. It is a framework that is used to build physical theories. In learning and understanding quantum mechanics, we will be learning something that actually doesn’t look like physics at all, but more like a set of abstract rules. My favourite statement of this fact is one that Scott Aaronson, a professor at MIT, makes in his book, Quantum Computing since Democritus:

“Basically, quantum mechanics is the operating system that other physical theories must run on as application software […]. But if quantum mechanics isn’t physics in the usual sense – if it’s not about matter, or energy, or waves, or particles – then what is it about? From my perspective, it’s about information and probabilities and observables, and how they relate to each other.”

This quote is so good, it is even used to sell printers in Australia, as seen in the short ad below.

When physicists talk about a quantum theory, they are referring to a physical theory that follows the framework of quantum mechanics. For example, quantum optics is a theory about the behaviour of light that is constructed according to the rules of quantum mechanics. This is also why it makes sense to talk about quantum computation: it is simply a theory of computation where computers follow the rules of quantum mechanics. Therefore, from now on, instead of talking about electrons and atoms, we will discuss a set of abstract rules that we can later apply to the context of interest.

Lesson 2: Ket notation. An essential starting point in quantum mechanics is the concept of the state of a system, which simply corresponds to one of its possible configurations. For example, when studying a Canadian one dollar coin, we assign one state to the coin showing the face of the queen (Heads) and another state to the coin showing a loonie (Tails). If the coin is in the configuration shown on the left in the image below, we say that it is in state “Tails”.


We can do the same for any system of interest. For example, we can assign a state to each of the faces of a dice, to the balance of your savings account, or to the energy of an electron in a hydrogen atom. Essentially, all we are really doing is assigning a label to the possible configurations we are considering.

In quantum mechanics we have a particular notation for the state of a system, which we call a “ket”. Although it may seem strange and pointless, it comes in handy when doing advanced calculations and it is a great way of making clear that we are referring to a quantum state. The way this works is that instead of writing a state in quotation marks – like I have done so far – we write inside of an uneven bracket. For example, instead of writing “Heads” or “Tails”, we write |Heads\rangle and |Tails\rangle. Similarly, when listing all the states of a dice we write:

|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle, |6\rangle.

A possible state of your savings account would be written as |+1,203\rangle . A cat could be in the state |Alive\rangle or the state |Dead\rangle. You get the idea.

I have introduced this notation for two reasons. First, I want to use the notation that cutting-edge researchers employ every day. Second, I want you to be able to recognize, just by glimpsing, that the equation written in a whiteboard has something to do with quantum mechanics.

Big Bang 

Lesson 3: Superposition. In the previous lesson, you were perhaps wondering what was so quantum anyway about writing |Heads> instead of “Heads”. If that crossed your mind, you are right! There’s nothing quantum about that, it’s just notation. Now I am going to tell you one of the two main features that makes quantum mechanics different. In the quantum world, there are more possibilities for the kinds of states that a system can be in. For example, the state:

|Heads\rangle + |Tails\rangle

is also a valid quantum state of a coin. We refer to such a state as being in a superposition of both states. In fact, the state:

|Heads\rangle - |Tails\rangle

is also a valid quantum state, although in a different kind of superposition. We’ll discuss this difference in another lesson. Similarly, the state:

|1\rangle + |2\rangle -|3\rangle -|4\rangle + |5\rangle + |6\rangle

is also a valid state of a quantum dice. In general, we can add and even subtract any set of states in any way we want, or even multiply them by any number, and end up with another valid state[1]. There are as many valid states as there are ways of combing them together by adding and multiplying by a number! For example, the following are also valid states of a quantum coin and a quantum dice, respectively:

-4 |Heads\rangle - 2|Tails\rangle

|2\rangle + 2|3\rangle -|4\rangle +0.5 |5\rangle

In quantum mechanics, there are so many more possibilities. Can you write down other interesting states of a quantum dice or coin?

In lesson 4, you’ll learn that we have to make an important adjustment to this rule, but for now, the important point is that there is an added richness in quantum mechanics in terms of the possible configurations of physical systems, since it is now possible for them to be in superposition.

What does it mean for a coin to be in the state |Heads\rangle + |Tails\rangle? That is a great question. The answer is… we don’t really know. People have come up with different proposals, from multiple universes to subjective views of reality, but none of them is entirely satisfactory. This is precisely the kind of situation that led Feynman to believe that no one understands quantum mechanics. But of course we understand! Quantum mechanics tells us that we can have superpositions, which are just sums of different states. This is not hard to grasp, is it? The difficulty comes when we ask questions regarding a connection of the theory with the natural world. These are very important and interesting questions that we should ask and try to answer and that I am personally very interested in. But strictly speaking, they are not questions about the theory. Remember, quantum mechanics is a framework, a set of rules that any physical theory must obey. Are these rules hard to understand? I don’t think so!

While the message of this lesson sinks in, take some time to admire this beautiful picture of the Milky Way galaxy.

milky way

Lesson 4: Measurements. Speaking of questions, what happens when we ask a quantum system what state it is in? In other words, what happens when we measure a system to determine its state? To provide an answer, we first have to make an adjustment to lesson 3. Remember I claimed that we could add up states in any way we wanted? Strictly speaking, this isn’t true: the state that we end up with must satisfy an additional property. It’s easiest to illustrate with an example. In quantum mechanics, instead of writing:

|Heads\rangle + |Tails\rangle

We must actually write:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

The numbers \frac{1}{\sqrt{2}} are referred to as the coefficients of the state. For example, what are the coefficients of the state:

\frac{3}{5}|Heads\rangle + \frac{4}{\sqrt{5}}|Tails\rangle ?

That’s right, the coefficients are \frac{3}{5} and \frac{4}{5}. Quantum mechanics requires that, when we square the coefficients and add them together, the result must always equal \(1\). In this case, we say that the state is normalized. For example, since:

(\frac{1}{\sqrt{2}})^2+(\frac{1}{\sqrt{2}})^2 = \frac{1}{2}+\frac{1}{2}=1

then our new state satisfies the rule. Similarly, the state:

is normalized because:


Fortunately, we can always normalize a state by simply re-scaling its coefficients appropriately, which is why I didn’t bother doing it in the previous lesson.

Good news for all the students who don’t like math: that’s the hardest math we’ll have to use learning quantum mechanics! 


Now, suppose that we have a quantum coin in a given state and we want to measure it. In quantum mechanics, a measurement is equivalent to asking a system: “In which of the following states are you in?” For example, in the case of a quantum coin, a valid measurement is to ask: “Are you in state |Heads\rangle or in state |Tails\rangle?” Quantum mechanics then tells us that the outcome of the measurement – i.e. the answer to our question – will definitely be either |Heads\rangle or |Tails\rangle and the probability that each outcome occurs is equal to the square of the corresponding coefficient.

Let’s look at some examples. Suppose that we have a quantum coin in the state:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

and we ask: “Are you in state |Heads\rangle or in state |Tails\rangle?”

Then with probability (\frac{1}{\sqrt{2}})^2=\frac{1}{2} the outcome will be |Heads\rangle and with probability (\frac{1}{\sqrt{2}})^2=\frac{1}{2} the outcome will be |Tails\rangle .

Similarly, if we have a quantum coin in the state:

\frac{1}{2}|Heads\rangle + \frac{\sqrt{3}}{2}|Tails\rangle

and ask the same question, with probability (\frac{1}{2})^2=\frac{1}{4} the outcome will be |Heads\rangle and with probability (\frac{\sqrt{3}}{2})^2=\frac{3}{4} the outcome will be |Tails\rangle .

Moreover, if we ask two questions in a row, then quantum mechanics tells us that we always get the same answer. In other words, the theory doesn’t contradict itself. We would be in trouble otherwise! In our first example, it was equally likely to obtain either of the outcomes. But after an outcome has occurred, let’s say |Heads\rangle , if we ask the same question we’ll always obtain |Heads\rangle as the answer, no matter how many times we ask.

This is a good place for a break, so before taking a look at some examples, let’s take a quick break to admire the beautiful building that hosts IQC and from which I wrote these words.


You may be wondering: how is this quantum coin any different than a regular coin? That’s an excellent question! Remember how quantum mechanics tells us that we can have a richer class of states thanks to superpositions? Well, in quantum mechanics we also have a richer class of measurements! For example, the question:

“Are you in state \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle or in state \frac{1}{\sqrt{2}}|Heads\rangle -\frac{1}{\sqrt{2}}|Tails\rangle ?”

is a valid measurement in quantum mechanics. This allows something truly unique to happen: A quantum coin in a given state can give probabilistic outcomes with respect to one measurement and deterministic outcomes with respect to another measurement. No regular coin can do this! For example, what happens if we have a coin in the state:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

and we ask the question:

“Are you in state \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle or in state \frac{1}{\sqrt{2}}|Heads\rangle -\frac{1}{\sqrt{2}}|Tails\rangle ?”

Well, with certainty, we are going to get \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle as the answer! This is completely different from what happened with our other question, where we obtained a completely random outcome. John Preskill, a professor at Caltech in the United States, refers to this ability to make different possible measurements by saying that “in the quantum case, there is more than one way to open the box.” Amazing, isn’t it?

The richer class of quantum states and the richer class of quantum measurements that arise because of superposition gives rise to many unique quantum properties – such as the uncertainty principle and entanglement – as well as important implications to the ways in which we can manipulate information, leading to exciting research fields such as quantum computing and quantum cryptography. This will be the topic of the last part of this series, in which we’ll use our new understanding of the rules of quantum mechanics to unravel their implications and potential technological applications.

Thank you for staying until the end of the lesson, I hope it was as much fun for you as it was for me!


[1] In fact, quantum mechanics allows us to use complex numbers in the superposition.


Anyone can understand quantum mechanics — Part 1


Pixar’s delightful movie Ratatouille – an animated film inspired by the world of French haute cuisine – features two characters with opposing views on cooking. On one hand we have Gusteau, a jolly and chubby chef with an optimistic message that he constantly repeats in his books and TV shows:


On the other hand there is Anton Ego, a famous food critic who doesn’t just like food, he loves it. And if he doesn’t love it, he doesn’t eat it (hence his skinny figure). Mr. Ego is of a very different opinion than chef Gusteau:


At the end of the movie (spoiler alert!), Anton Ego has a change of heart, which he expresses beautifully in one of the greatest reviews ever.

To me, the answer to their dilemma is obvious: of course anyone can cook, cooking is easy – even I can do it! It definitely takes practice, with a few burnt dishes along the way, but it’s not like there is an insurmountable barrier only a few chosen ones can overcome. In fact, out of plain necessity, billions of people around the world cook their own food every day. However, it is also clear that we all cook with different degrees of skill and not anyone can do it as well as the best chefs in the world. For instance, I doubt that I could ever prepare this dish, “Lily bulbs, distillation of finger lime and lily and apple blossoms” from the restaurant Alinea in Chicago:


Looks amazing! How do they do this?

So what does all of this have to do with quantum mechanics?

Simple: I believe that the Anton Egos of the scientific world have been spreading the message that quantum mechanics is a mind-boggling, counter-intuitive, and spooky theory that not even the greatest geniuses can understand. The truth, however, is that quantum mechanics is as easy to grasp as any other theory in science and its basic principles are something that anyone can understand; just like cooking.

Why would anyone spread the message that quantum mechanics is so difficult to understand?

A hundred years ago, when quantum mechanics was first emerging as a physical theory, everyone was extremely confused. Researchers had been trained following a scientific tradition that had been brilliantly successful at explaining the natural world and which gave a very compelling picture of what the universe was really like. In particular, scientists had been taught to view the universe as composed of elementary particles whose properties could be known and whose behaviour could, in principle, be predicted perfectly. Just try and imagine how difficult it was for them to deal with a new theory that was turning this picture of the universe upside down, a theory which claimed that the properties of physical systems couldn’t always be known perfectly and their behaviour could only be predicted probabilistically! The situation was not unlike what happened during Dagen H, the day when Sweden changed from driving on the left-hand side of the road to the right. Chaos, confusion, and disputes were inevitable.


Kungsgatan in Stockholm on Dagen H, September 3, 1967

As a consequence of this clash with pre-existing views came the first examples of renowned physicists boasting about the difficulty of understanding quantum mechanics. In fact, at the time, there were many scientists who were manifestly arguing that the theory had to be wrong or incomplete. In retrospect, this comes as no surprise; it is to be as expected as traffic jams in Stockholm on September 3, 1967. The issue is that this confusion continues to propagate in many introductory courses and most tragically, in countless books and articles for the general public. As an example, the following is a recurring quote by Niels Bohr which I find to be the most irritating:


If Bohr is right, then either we’ll be shocked when learning one of the most important and fundamental truths about the universe – as opposed to enlightened and happy – or if we aren’t shocked, we’ll have to come to terms with the fact that we are not smart enough to comprehend it. A no-win situation. Thanks Niels.

This was the first generation of quantum physicists, people who were around at the time when the theory was actually proposed and built from scratch. With them came the first wave of confusion, caused by the clash between their pre-existing worldview and the basic principles of quantum mechanics. Their bewilderment is understandable and to be expected, but not something we should be teaching in the 21st century. Think about it, why would anyone keep bringing up this quote by Bohr? What are they trying to achieve? Nothing good in my opinion.

After quantum mechanics was developed into a complete and well-defined mathematical theory, it quickly begun to be successfully applied to many areas of physics. This lead to new technologies such as lasers and transistors, as well as to a deeper understanding of the properties of matter and elementary particles. This was the second generation of quantum physicists, the people that grew up with the theory and learnt about it in university, people who even had textbooks to follow with exercises and solutions in the back. With them came the second wave of confusion. Their generation did not have doubts concerning the validity of the theory, which had passed every single empirical test with flying colours. The issue was that nobody could pin down what kind of universe is ruled by the laws of quantum mechanics! They knew the rules, but could not provide a coherent picture of the world to go with it. To no surprise, once again came famous figures in science telling us that the quest for understanding quantum mechanics was destined for failure, for not even the greatest geniuses could emerge victorious. The most annoying of such claims is this quote by Feynman:


So, the person who won a Nobel Prize for developing quantum electrodynamics and wrote a best-selling series of physics textbooks is saying that nobody understands quantum mechanics? No way! Why did they give him a prize then? Why are people buying his textbooks? What Feynman meant was that there remained some profound foundational questions about quantum mechanics for which nobody had been able to provide satisfactory answers. To him, this meant that the theory was not completely understood. But of course it was understood, he was teaching it to hundreds of students!

Why do we keep quoting Feynman on this? How are we supposed to get people excited about learning quantum mechanics when we tell them from the start that nobody can understand it?

Thankfully, like Bob Dylan said, “the times they are a changin.” Now it’s time for us, the third generation of quantum scientists, to give quantum mechanics the reputation it deserves: that of a beautiful and simple theory that should be understood by as many people as possible. For example, at the Institute for Quantum Computing (where I did my PhD), they are making great efforts to share our research and teach quantum theory to the world. Every year they host around 40 high school students from Canada and the world and teach them the basics of quantum mechanics in the Quantum Cryptography School for Young Students. IQC has also recently started a training program called Teaching Quantum Technology, aimed for teachers who are interested in introducing the ideas behind quantum mechanics and their application to technology in their classrooms. Recently, they even created a video game: Quantum Cats! (I helped 0.1% to develop it.) This is only an example of a worldwide trend to spread our knowledge of quantum mechanics, a movement that is guided by the conviction that Bohr and Feynman were wrong: anyone can understand quantum mechanics.

If you are interested in being one of the growing number of people that have added quantum theory to their database of knowledge, please join me for part 2 of this series, where I will teach you the basics of this most splendid theory. No matter your age, your background or your preferences, I am sure that these are ideas that you can grasp and that you will enjoy learning. I will be expecting you!