# The eerie phenomenon of quantum entaglement

In this article, I will discuss the weird phenomenon called quantum entanglement, to which Albert Einstein referred as "spooky action at a distance".

After reading this article, you will understand what is so spooky about this and learn how we can show that quantum entanglement is something that can not be explained without quantum mechanics. But first a little bit of history. When quantum mechanics was being developed as physical theory of the small (think electrons, protons etcetera), many physicists could not accept it, because of the counter-intuitive nature of some of its predictions. One of these physicists was none other than Albert Einstein himself, till his death he simply could not accept quantum mechanics as a viable theory for our universe. He made his main objection in a thought experiment now known as the EPR paradox [EPR35], where $\text{EPR}"$ stands for Einstein-Podolsky-Rosen, named after the physicists who came up with it. In this thought experiment, they found that quantum mechanics predicts a phenomenon called quantum entanglement.

Exercise: What quantum state does a fair coin represent in the analogy described above?

# The thought experiment

In their article, Einstein and his colleagues described the following thought experiment.

Consider a pair of particles, say electrons for concreteness, which we prepare in a certain way in the lab: the particles become $\text{entangled}"$. How this is done, is not important for us, but experimentalists can do this.

After being prepared, the particles are separated. Since this is a thought experiment, assume that one electron is with you here on Earth and I go to Mars together with the other electron. Now that we are separated with our electrons, we measure the electrons. The measurement device then outputs a $0$ or $1$ with probability $\frac{1}{2}$, like a coin flip.

It is very natural to think that what I measure and what you measure are independent of each other, since we are so far from each other. But according to the rules of quantum mechanics, the way the electrons were prepared makes it so that whatever my measurement device outputs, it is going to be the same as what your device outputs!

Figure 1: An artists representation of two entangled states shared between two diamonds. Taken from qutech.nl [1].

If you think for a while about how this could be possible, then the following two options might occur to you.

First option: When the electrons were prepared, they carried a shared "hidden variable" with them.

Second option: The electrons somehow communicate after they get measured (the first electron is measured and outputs, say, spin-up and instantaneously sends this information to the other electron so that it can also say spin-up after measurement).

Let's go through both options. Imagine that the electrons are intelligent (I know, this sounds silly) and are able to flip coins! In this way, option 1 can be explained as follows. When the electrons were together in the lab, they secretly flipped a coin and remembered the outcome of this coin flip. As explained above, the electrons then get separated from each other and are measured. When the electrons get measured, they output the outcome of the coin flip they did earlier. In this way, the experimentalists observe that they always output the same answer every time they do this experiment. What about option 2?

According to Einstein's theory of special relativity, information can not travel faster than the speed of light. So if I measure my electron on Mars and the sneaky electron sends this measurement outcome (0 or 1) to your electron on Earth, it will take a few minutes for this information to arrive. But according to quantum mechanics, it does not matter when you measure your electron. Even if we measure exactly at the same time, we will always get the same output! It is as if this information is traveling faster than the speed of light. This is exactly the reason why Einstein favored the first option. So let us dismiss option 2 and focus on option 1.

Quantum mechanics does not say anything about this hidden variable that we just discussed. So if there really is such a hidden variable, then quantum mechanics can not be complete as a theory that explains the tiny world of electrons and other particles. This is the reason why Einstein could not accept quantum mechanics. If only there was a way to experimentally test if the electrons are  flipping coins...

In 1964 (9 years after Einstein died) John Bell had a ground-breaking insight [Bel66]. He found a way to determine if there is such a thing as a hidden variable. Using his ideas, we can test if the electrons are secretly flipping coins! In order to understand his ingenious idea we need the framework of non-local games [CHTW04], a very nice mathematical framework in which we can study this eerie phenomenon of quantum entanglement.

## Non-local games

So what is a non-local game? There are two players, Alice and Bob, and a verifier. The verifier picks two questions from a set of questions randomly and sends one to Alice and the other to Bob separately. Alice does not know what the verifier asks Bob and vice versa. Alice and Bob then have to answer from some set of answers. The verifier will then examine the answers of both players and decide if they win or not. He verifies if they answered correctly together. The question set, answer set and the winning condition are all known to the players before the game starts. Alice and Bob can think of a strategy before the game starts, but once the game starts, they can no longer communicate.

The CHSH game [CHSH69] is a very famous non-local game and this will serve as our example. The game is named after its inventors Claude, Horne, Shimony and Holt. The numbers 0 and 1 are the $\text{questions}"$. Alice and Bob have to answer with either 0 or 1 given a question. So suppose Alice gets $i$ and Bob gets $j$ as a question, which can be either 0 or 1. They return answers $a$ and $b$ which can also be either 0 or 1.

The players win if $a\oplus b = i\cdot j$. Here $\oplus$ is called the XOR operation on bits. It is given by the following table.

This operation can be phrased in words as follows: if $a$ and $b$ are the same, then $a\oplus b = 0$ and if $a$ and $b$ are different, then $a\oplus b = 1$. And $i\cdot j$ is the usual product of numbers.

## Allowed strategies

There are two kinds of strategies that the players are allowed to use. The first type is called a classical strategy. In this case, the players discuss before the game starts what to answer given a question. They are also allowed to use a shared source of randomness. For example, they could have access to random coin flips they both can see. Deterministic strategies are classical strategies but without the use of shared randomness, and it turns out that you can always get the best winning probability using a deterministic strategy. An example strategy is as follows: if Alice gets 0 as a question, she will answer with 0 and if she gets 1 as a question, she will answer with 1. Bob will always return 0 as an answer.

The second type of strategy is called a quantum strategy. In this case the players are allowed to use quantum entanglement to try and beat the game. Alice and Bob first create entanglement, say, between two electrons in the lab as described earlier. Alice then takes one electron with her and Bob the other one. They decide how to measure their electrons when asked a certain question.

The measurement outcomes are 0 or 1 and this is going to be their answer to the referee. I will not go in to more detail about quantum strategies, as it is mathematically a little bit involved. The figure on the left illustrates the situations just described.

If Alice and Bob could talk to each other, they could easily win the game.

Exercise: How?

### Classical strategies

What strategy can they devise if they are not allowed to communicate as soon as they get their inputs? Let us look at the following table, it consists of possible inputs for both Alice and Bob and variables representing their answer.

The question is: can we fill in the table for values of $a_0, a_1, b_0, b_1$ such that the winning condition, $a\oplus b=i\cdot j$, is always satisfied? In other words, does there exist a perfect classical strategy? If we add together all the equations, we can see that each variable $a_i$ and $b_i$ occurs twice. When taking the XOR of the same variable we always get 0, i.e. $a\oplus a = 0$. But the right-hand side of the equation is 1 since we are adding $0\oplus0\oplus0\oplus 1 = 1$. Assuming there exists a perfect classical strategy, we conclude
that $0 = 1$! This is a contradiction, so there can not be a perfect classical strategy. The best Alice and Bob can do, is win with probability $\frac{3}{4}$.

Exercise: Can you give a strategy for Alice and Bob that wins with probability $\frac{3}{4}$ ?

### A quantum strategy does better

However, if the players are allowed to use quantum entanglement, they can devise a strategy such that they win the game with a probability of $0.85$! The conclusion is that in theory, quantum entanglement can give an advantage.

#### Bell's idea

Now that we know what a non-local game is and all of its ingredients, we can discuss Bell's idea. Remember that we wanted to know if the electrons in our thought experiment were secretly flipping coins or that there is really something else going on. Something that only quantum mechanics can explain.

In terms of non-local games, two electrons that give answers to a measurement using coin flips corresponds exactly with Alice and Bob playing a non-local game using a classical strategy using shared randomness. Here comes the mind twist: if Alice and Bob can win a non-local game using a quantum strategy by using entanglement between two electrons with higher probability than they would if they only used classical strategies, then that means that the electrons are not secretly flipping coins! It might take some time to digest this. How do we $\text{play}"$ such a game in the real world then? In reality, experimentalists first create entanglement between two electrons and separate the two systems. The verifier then chooses, just as before, two numbers at random which can be 0 or 1. These numbers are sent to the two systems. The optimal strategy, mentioned above, tells the experimentalists how to measure their electrons depending on what they got from the verifier. The measurement then outputs a number and this is what the experimentalists tell the verifier. The verifier then checks if they are correct. In theory, the measurement outcomes will be correct in $0.85$ fraction of the time that they perform this experiment.

#### Quantum entanglement is real

Indeed, in the 80's it was the first time that this experiment was realized by Alain Aspect, Philippe Grangier, and Gerard Roger [AGR81] and it was shown that the probability of winning is larger than $\frac{3}{4}$. Of course, researchers did not stop there. These games have been played many more times over the years. And with physical equipment getting better, we can be more certain that quantum mechanics as a physical theory of the small is not incomplete as Einstein thought and that quantum entanglement is a phenomenon that we can understand using quantum mechanics.

The story does not stop here. If we can harness the power of quantum entanglement, many things become possible. In particular, researchers are working on a $\text{quantum internet}"$ which, without entanglement, would not be possible, see for example this Network Pages article. Another field where the rules of quantum mechanics, and in particular quantum entanglement, can be used is in algorithms. In the field of quantum computing, see for example [NC02], it is the aim to develop quantum algorithms for certain computational problems where quantum mechanics might give an edge over any classical algorithm. This might be the topic for a future Network Pages blog post!

[AGR81] Alain Aspect, Philippe Grangier, and Gerard Roger. Experimental tests of realistic local theories via bell's theorem.
[Bel66] John S. Bell. On the problem of hidden variables in quantum mechanics.
[CHSH69] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories.
[CHTW04] Richard Cleve, Peter Hoyer, Benjamin Toner, and John Watrous. Consequences and limits of nonlocal strategies.
[EPR35] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete?
[NC02] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information.