Before we begin, HAVE YOU WATCHED THE VIDEO “ANYONE CAN QUANTUM”??? Paul Rudd, Keanu Reeves, Stephen Hawking, Quantum Chess, Quantum Physics for Babies, and even tardigrades: this video has it all!

Made by our colleagues from the Institute for Quantum Information and Matter in Caltech, this clip has masterfully shared with almost **two million people** all around the world the same message that I have been trying to spread with these blog posts: anyone can understand quantum mechanics! In their video, Keanu Reeves tells us that “Paul Rudd changed the world by showing the world that anyone can grapple with the concepts of quantum mechanics. It sparked an era of invention and ingenuity the likes of which humanity had never seen.” There is a lot of truth in that: when everyone believes they can understand nature at its most fundamental level, we can accomplish amazing things.

Of course, it’s one thing to *claim *that people can understand quantum mechanics, but it’s something else entirely to help people actually do it. That’s the job I started with my previous posts and that I am going to continue today. Let’s go!

In the previous installment of this series, we learned the basic postulates of quantum mechanics. In other words, we learned what quantum mechanics *is.* In this final part of the series, we are going to shift gears and study what quantum mechanics *implies *about our universe. What is possible in a quantum world that can’t be done in a classical one? What is new, what is different? These are important questions and today we’ll learn some of the answers!

My job, for instance, could be loosely described as studying the implications of quantum mechanics for communication, cryptography and thermodynamics. In particular, in this lesson we’ll learn about two important aspects of quantum mechanics that are not present in classical theories: *Heisenberg’s uncertainty principle and entanglement*. Uhu!

**Lesson 1: The uncertainty principle**

In our previous lesson, we used a quantum coin as an example of a quantum system, whose state could be \(|Heads\rangle\), \(|Tails\rangle\) or any superposition of these two states. Today, we are going to be more general and instead we are going to think of a system with two possible configurations which we call \(|0\rangle\) and \(|1\rangle\). Using this notation is great because it’s shorter to write (which is always appreciated) and because it is more general: we don’t really need to be talking about a quantum coin, it could be any system with two degrees of freedom. The word we use for such an object is a *qubit, *in analogy with a classical bit, which is any system that can be in states 0 or 1.

Remember that in quantum mechanics we can have superpositions, so we are also going to define two other important states of a qubit

\[|+\rangle=\frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle \]

\[|-\rangle=\frac{1}{\sqrt{2}}|0\rangle - \frac{1}{\sqrt{2}}|1\rangle \]

Notice that the states \(|+\rangle \) and \(|-\rangle \) are both an equal superposition of the states \(|0\rangle \) and \(|1\rangle \). Notice also that the states \(|0\rangle \) and \(|1\rangle \) are equal superpositions of \(|+\rangle \) and \(|-\rangle \) since we can write

\[|0\rangle=\frac{1}{\sqrt{2}}|+\rangle + \frac{1}{\sqrt{2}}|-\rangle \]

\[|1\rangle=\frac{1}{\sqrt{2}}|+\rangle - \frac{1}{\sqrt{2}}|-\rangle \]

Do you remember how to define a measurement in quantum mechanics from the previous lesson? The only thing we have to do is to ask systems what states they are in. In this lesson, we are going to focus on two special measurements of a qubit. We’ll call the question

“Are you in state \(|0\rangle\) or \(|1\rangle \) ?”

a *Z measurement.* Similarly, we’ll call the question

“Are you in state \(\|+\rangle\) or \(|-\rangle\) ?”

an *X measurement*. Why do we call them that? Well, there’s a relatively complicated historical reason behind it, but this is the terminology that scientists use and I want you to be familiar with these terms.

Now let’s suppose that we have a qubit in the state \(|0\rangle\) and we want to measure it. If we make a Z measurement, we know for sure that the outcome will be “I’m in state \(|0\rangle\)”. However, because of the laws of quantum mechanics that we learnt last time, if we make an X measurement, half of the time we’ll obtain the outcome “I’m in state \(|+\rangle\)” and the other half we’ll get the outcome “I’m in state \(|-\rangle\)”. In other words, for this state, we don’t have any uncertainty about the outcome of a Z measurement, but we have maximum uncertainty about the outcome of an X measurement. See where I’m going?

What happens if instead we start with a qubit in the state \(|+\rangle\)? You guessed it, the situation is reversed! In this case, we don’t have any uncertainty about the outcome of an X measurement, but we have maximum uncertainty about the outcome of a Z measurement! It turns out that no matter what state we start with, *there will always be some uncertainty in at least one of these two measurements*. That is Heisenberg’s uncertainty principle.

More precisely, the uncertainty principle states that for virtually any two measurements we can make on any system – let’s call them measurement A and measurement B – it holds that

**Uncertainty(A)+Uncertainty(B)>0.**

In other words, no matter what state the system is in, there exist pairs of measurements whose outcomes cannot *both* be predicted perfectly. This never happens classically! In a classical world, if we know the state of a system perfectly, in principle we can predict everything about its future behaviour, including the results of any two measurements. But in a quantum world, there is a fundamental limitation to our ability to predict the outcomes of measurements: most of the time, there will always be some uncertainty about which outcomes we’ll see. The only exception to this rule occurs when both A and B are said to *commute, *but most pairs of measurements don’t have that property.

Many of you are probably thinking, “Wait, didn’t the uncertainty principle have something to do with the position and momentum of a particle?”

Well, the uncertainty principle applies to measurements of position and momentum as well: we can never predict the outcome of both measurements perfectly. In other words, in our universe, for any measurement X of the position of a particle and any measurement P of its momentum, it holds that

**Uncertainty(X)+Uncertainty(P)>0.**

The uncertainty principle tells us something very deep about our ability to obtain information from physical systems. In many ways, it sets a fundamental limit to our capability to make predictions and to perform precise measurements. This has HUGE implications. To name a few, the uncertainty principle is the reason why quantum states cannot be cloned, why empty space is not really empty, and why quantum cryptography is possible. That’s the beauty of our quantum world!

Before our next lesson, you can take a break and admire this picture that my wife took of the Gardens by the Bay in Singapore, the city where we now live.

**Lesson 2: Entanglement**

So far in our discussion of quantum mechanics we have focused on single systems: a single quantum coin, a single quantum die, a single qubit. But what happens if we combine systems together? In particular, what happens if we have two qubits instead of one?

The first thing we have to understand is how to *represent* the states of two qubits. Turns out that all we have to do is to “stick them together”. If one qubit is in state \(|0\rangle\) and the other is in state \(|1\rangle\), then we represent the joint state of both qubits as \(|0\rangle |1\rangle\). Easy! Mathematicians call this operation “taking the tensor product”, I prefer to use the term “sticking them together”: it gets the point across.

Other examples of possible states of two qubits are

\[|1\rangle |0\rangle\]

\[|+\rangle |1\rangle\]

\[|-\rangle |+\rangle\]

\[(\frac{3}{5}|0\rangle +\frac{4}{5}|1\rangle) |0\rangle\]

You get the idea. Notice that in each of these examples, it is straightforward to identify what state each of the two individual qubits is in. For instance, for the state \(|-\rangle |+\rangle\), it is clear that the first qubit is in state \(|-\rangle\) and the second qubit is in state \(|+\rangle\).

Now comes the interesting part. Remember that in quantum mechanics we can have *superpositions *of different states. Hopefully many of you are already realizing that much of the magic of the quantum world comes solely from superposition: it is one of the defining properties that makes quantum mechanics such a beautiful and rich theory. For example, in quantum mechanics, a system of two qubits can be in the state

\[\frac{1}{\sqrt{2}}(|0\rangle |1\rangle + |1\rangle |0\rangle) \]

Does this state look special to you? If not, then let me ask you a couple of questions: what state is the first qubit in? What state is the second qubit in? Think about it for a while.

So, what’s the answer? That’s right: they don’t have a definite state! In fact, if we perform *any *measurement in either of the two qubits, we will always get a completely random outcome. Thus, this peculiar state has the intriguing property that even though we know the state of *both *qubits perfectly, we are completely ignorant of the state of each *individual *qubit. Mind-blowing isn’t it?

Any state that *cannot *be written in the form \(|state1\rangle|state2\rangle\) is called *entangled*, where \(|state1\rangle\) is some state of the first qubit and \(|state2\rangle\) is some state of the second qubit. You can check for yourself that indeed the state

\[\frac{1}{\sqrt{2}}(|0\rangle |1\rangle + |1\rangle |0\rangle) \]

which from now on we’ll call \(|\Psi\rangle\), cannot be written in this form and is therefore an entangled state.

The Centre for Quantum Technologies (CQT), where I now work as a research fellow, organized a mini-competition last year to coin a new way of referring to entanglement to replace the popular “spooky action at a distance”, which I dislike (more on that in a few minutes). The winner entry was “Mutual existence”, which was chosen by writer George Musser and CQT professor Christian Kurtsiefer. You can read more about it and other entries here. In Musser’s words “I like 'mutual existence' because it captures the principle that entangled particles behave as a single unified system, with global properties that do not reside on either particle, or even derive from them.” Now you know what he means! The joint state of two entangled systems is perfectly defined, but in such a way that their individual states are not. Beautiful!

Now what happens if we measure one of the qubits in an entangled state? Well, we know we’ll get *some *outcome, but as you might have guessed, because the state of each individual qubit is not well defined, no matter what measurement we make, we’ll always obtain a random answer. If we measured the first qubit of state \(|\Psi\rangle\) by asking “are you in state \(|0\rangle\) or in state \(|1\rangle\)?” we’ll obtain each possible answer with 50% probability. But notice something amazing: because quantum mechanics always gives consistent answers, if we then measure the second qubit *we know what outcome we’ll obtain*! If the outcome of the measurement of qubit 1 was “I’m in state \(|0\rangle\)” then for sure we’ll obtain outcome “I’m in state \(|1\rangle\)” when we measure qubit 2, since \(|\Psi\rangle\) was an equal superposition of \(|0\rangle|1\rangle\) and \(|1\rangle|0\rangle\). Moreover, this is true no matter how far apart the qubits are from each other.

Many people were frightened by this realization: the state of qubit 2 is initially not well-defined, but as soon as we measure qubit 1, we immediately know the state of qubit 2. This is what led Einstein to call this effect “spooky action at a distance”. But as you’ll see, it’s not spooky and it’s not action at a distance.

Following the argument of the great John Stewart Bell, imagine there is a person that always wears socks of different colours. In Bell’s case, this was his friend, Reinhold Bertlmann. On a given day, it was impossible to predict what sock he would wear on each foot. However, if you got a glimpse at *one* of his socks then you immediately knew that the other sock must be of a different colour. Sounds familiar?

So you see, there is nothing quantum about objects being correlated in this way: even if their states are uncertain, their shared properties may allow us to make inferences about one of them from knowledge of the other. Here’s what’s quantum about entangled states: this powerful correlation remains no matter what measurements we make!

Once again, in quantum mechanics, we have superpositions, so we can ask a richer class of questions. In the case of the entangled state \(|\Psi\rangle\) , we could ask the first qubit “Are you in state \(|+\rangle\) or in state \(|-\rangle\).” You can check for yourselves (or trust me on this) that we can equivalently write \(|\Psi\rangle\) as

\[|\Psi\rangle=\frac{1}{\sqrt{2}}(|+\rangle |+\rangle - |-\rangle |-\rangle)\]

so now we know that the outcome of the same measurement on qubit 2 will always be the same as for qubit 1. This correlation between several different measurements is not possible to achieve classically: entangled states have much stronger correlations. That’s the reason that my personal entry for the mini competition was this:

Read individual Tweet on Twitter

Quantum correlations are stronger than classical ones and they lead to a myriad of applications, like randomness generation, quantum cryptography and quantum teleportation. Perhaps most importantly, as shown by Bell in the 1970s, the properties of entangled states have taught us that we cannot understand the world as being one in which the outcomes of all events have been pre-established and where signals cannot travel faster than light: at least one of these two principles does not hold in our universe.

I hope you have enjoyed this trip across the quantum world. My honest hope is to have given you an understanding of the basic concepts of quantum mechanics and, most importantly, to have ignited a desire to learn more about this most beautiful of theories.

## Comments

Alex Wozniakowski

Wed, 2016-04-27 12:24

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This was an interesting read. After mentioning Bertlmann's socks you measured the Bell state, asking if the first qubit is in the state plus or plus. I propose changing the latter to minus.

Jodi Szimanski

Fri, 2016-04-29 08:26

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Thank you for noticing that Alex! We've corrected it.