# Simple quantum games reveal the ultimate complexity of the universe

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### A game for two can tell if the universe has an infinite amount of difficulty

*How many independent properties does the universe have? A simple game can answer this question.*

One of the greatest and most basic questions in physics concerns the number of ways to set up matter in the Universe. If we take matter and regroup it, then regroup again, and again - will we exhaust all possible configurations, or can these permutations be done indefinitely?

This is unknown to physicists, but in the absence of certainty they make assumptions. And these assumptions vary depending on the field of physics. In one field, physicists assume a finite number of configurations. In the other, the infinite. It is still impossible to say which of them is right.

But over the past couple of years, one group of mathematicians and computer scientists has been creating games that theoretically can solve this problem. In games two players participate, isolated from each other. Players ask questions, and win if their answers are agreed in a certain way. The number of wins is related to the number of different ways to configure the universe.

“There is a philosophical question: is the finite or infinite number of dimensions of the universe?” Said Henry Yuyen , a theoretical computer science specialist at the University of Toronto. "People think it's impossible to verify this, but one of the possible ways to resolve the issue is to use a game invented by William."

Yuen talks about William Slofstra, math from the University of Waterloo. In 2016, Slofstra invented a game for two players assigning values to variables in hundreds of simple equations. Under normal conditions, even the most skilled players can lose. But Slofstra proved that if you give them access to an infinite amount of unusual resources - entangled quantum particles - they can always win.

Other researchers have since corrected the result of Slofstra. They proved that in order to reach the same conclusion, one does not need to play a game with hundreds of questions. In 2017, three researchers proved that there are games of only five questions that can be won in 100% of cases if the player has access to an unlimited number of entangled particles.

All of these games are based on games invented more than 50 years ago by physicist John Stuart Bell. Bell developed games to test one of the strangest hypotheses put forward by quantum mechanics about the physical world. Half a century later, his ideas may prove useful not only for this.

## Magic squares

Bell came up with "non-local" games that require players to be at a great distance from each other, without the ability to communicate. Each player answers a question. Players win or lose depending on the compatibility of their answers.

One such game is the magic square. Players Alice and Bob draw a 3x3 grid of squares. The judge asks Alice to fill out one row in the grid — say, the second — by writing 1 or 0 in each cell so that the sum of the numbers in the row is odd. The judge then asks Bob to fill out one of the columns so that the amount is even. Alice and Bob win if they write the same number at the intersection of their row and column.

The catch is this: Alice and Bob do not know which line or column the judge asked their opponent to fill. “Such a game would be trivial if the players could communicate,” said Richard Cleve, a student of quantum computing at the University of Waterloo. “But the fact that Alice does not know what they asked to do Bob, and vice versa, means that the game is becoming more difficult.”

It seems that in a game with a magic square and other similar games there is no way to win in 100% of cases. Indeed, in the world described by classical physics, Alice and Bob can reach a maximum of 89%.

However, quantum mechanics - in particular, the strange phenomenon of "entanglement" - allows Alice and Bob to improve the result.

In quantum mechanics, the properties of fundamental particles, for example, electrons, do not exist until the moment of measurement. Imagine that an electron moves rapidly around a circle. To determine his location, we take a measurement. But before the measurement, the electron does not have a specific location. It is characterized by a mathematical formula that expresses the probability of finding it in a particular place.

When two particles are entangled, the complex amplitudes of the probabilities describing their properties are intertwined. Imagine two electrons entangled in such a way that if the measurement determines the location of one of them at a certain place in the circle, then the other will necessarily be at the opposite point. This relationship of the two electrons is preserved, and when they are close, and when they are separated over many light years. Even at such a distance, if you measure the location of one electron, the location of the other will become known immediately, even without a causal relationship between them.

This phenomenon seems absurd, because in our non-quantum experience there is nothing that would indicate such a possibility. Albert Einstein ridiculed the confusion with the famous phrase “frightening long-range action”, and for years claimed that this could not be.

To implement the quantum strategy in a game with a magic square, Alice and Bob take one of the entangled particles. To determine which numbers to write, they measure the properties of their particle - much like they would roll cubes connected to each other to select answers.

*John Stuart Bell, who invented non-local games*

Bell calculated, and many subsequent experiments showed that, using strange quantum particle correlations, players in such games can coordinate their answers much more accurately, and win more often than in 89% of cases.

Bell came up with non-local games as a way to prove that entanglement is real, and our classic view of the world is incomplete - and at that time such a conclusion was easy to draw. “Bell came up with an experiment that could be done in the lab,” Cleve said. If we manage to register a percentage of success that exceeds the expected, it will become clear that the players are using some features of the physical world that are not explained by classical physics.

The work done by Slofstroy and others is similar in strategy, but different in scale. They showed that Bell's games not only prove the reality of entanglement, but some of them can prove something more - for example, the existence of a limit on the number of configurations that the Universe can accept.

## More confusion

In 2016, Slofstra proposed a new non-local game, in which two players play, giving answers to simple questions. To win, they need to give answers, in a certain way connected with each other, as in a game with a magic square.

Imagine, for example, a game for two players, Alice and Bob, who need to match socks from their dressers. Each player must choose one sock, not knowing which sock the other chose. Players cannot agree on a choice in advance. If their socks come from the same pair, they win.

Given this uncertainty, it is not known which socks Alice and Bob should choose - at least in the classical world. But if they can use entangled particles, their chances of pairing up will increase. Based on the choice of the color of the sock on the measurement results of one pair of entangled particles, they can coordinate the selection of this one attribute of the sock.

However, they still have to guess about the other attributes - a woolen sock or a cotton sock, up to the ankle or to the middle of the calf. But, using additional intricate particles, they can access more dimensions. They can use one set to correlate the choice of material, the other to choose the length of the toe. As a result, due to the ability to coordinate the selection of many attributes, they are more likely to choose socks from one pair.

“More sophisticated systems allow you to make more consistent measurements, which allows you to coordinate actions when performing more complex tasks,” said Slofstra.

But in the game of Slofstra, questions do not apply to socks. They relate to equations such as a + b + c and b + c + d. Alice can assign any variable a value of 1 or 0 (and the value of each variable will remain the same for all equations). As a result, its equations in total will give a certain value.

Bob is given one of Alice's variables, for example, b, and is asked to assign her a value of 0 or 1. Players win if both assign one value to this variable.

If you were playing this game with a friend, you could not constantly win. But with the help of a pair of entangled particles, the gain would become more permanent, as in the example with socks.

It was interesting to Slofstra to understand whether there was a quantity of entangled particles, beyond which the probability of a team winning ceases to grow. Perhaps the players could build an optimal strategy, having on hand five pairs of entangled particles, or 500 pairs. “We were hoping we could say: for optimal play, it takes so much confusion,” said Slofstra. “But it turned out that this is not so.”

He found that adding extra entangled particles always increases the chance of winning. And if you could use an infinite number of entangled particles, you would be able to play this game perfectly, winning 100% of the time. With socks, this obviously does not work out - someday all the features of socks will end. But, as the game of Slofstra showed, the Universe can be much more complicated than a box with socks.

## Is the universe infinite?

The result of Slofstra shocked scientists. Eleven days after the appearance of this work, computer science specialist Scott Aaronson wrote that the result raises “a question of almost metaphysical importance: namely, what experiments in principle can show whether the Universe is discrete or continuous?”

Aaronson wrote about various states that can to accept the Universe, where the “state" is a certain configuration of all its matter. Each physical system has a space of states, or a list of all the various states that it can accept.

*William Slofstra, mathematician at the University of Waterloo*

Researchers talk about a certain number of measurements in the state space, reflecting the number of independent characteristics that can be configured in the system. For example, even the box with socks has a state space. Each sock can be described by color, length, material and wear. Then the state space of the box with socks has four dimensions.

The difficult question about the physical world is this: is there a limit to the size of the space of states of the Universe (or any physical system). If there is a limit, then it doesn’t matter how big and complex the physical system will be, it can only be configured in a finite number of ways. “The question is whether physics allows physical systems to exist with an infinite number of properties independent of each other, which can, in principle, be observed,” said Thomas Widick , an IT specialist at the California Institute of Technology.

So far, physicists have not decided on the answer. Moreover, there are two opposing points of view.

On the one hand, students in an introductory course in quantum mechanics are taught to think in terms of state spaces with an infinite number of dimensions. By simulating the location of an electron moving in a circle, they assign probability to each point in the circle. Since there are an infinite number of points, the state space describing the location of the electron will have an infinite number of dimensions.

“To describe the system, we need a parameter for every possible electron location,” Yuyen said. - There are infinitely many points, so we need an infinitely many parameters. Even in a one-dimensional space (circle), the state space of a particle has an infinite number of dimensions. ”

But perhaps the idea of an infinite dimension space does not make sense. In the 1970s, physicists Jacob Beckenstein and Stephen Hawking calculated that a black hole is the most complex physical system in the Universe, but even its state can be described by a large but finite number of parameters - approximately 10

^{69}bits of information per square meter of its event horizon. This number, the Beckenstein limit , suggests that if a black hole does not require a state space with an infinite number of dimensions, then nothing else is needed either.

These competing concepts of state spaces reflect fundamentally different views on the nature of physical reality. If state spaces have a finite number of dimensions, then on the smallest scale nature should be pixelated. But if electrons require state spaces with an infinite number of dimensions, physical reality is intrinsically continuous even at the smallest resolution.

So what is true? Physicists have not yet given an answer, but the game of Slofstra, in principle, can provide it. Slofstra's work offers a way to make a distinction: play a game that can be won at 100% only if the Universe allows state spaces with an infinite number of dimensions to exist. If players win every time, this means that they will take advantage of correlations that can only occur when measuring physical systems with an infinite number of independently adjustable parameters.

“He offers such an experiment that if it can be implemented, then we can conclude that the system that provides the observed statistics must have an infinite number of degrees of freedom,” said Vidik.

However, there are certain obstacles to the implementation of the Slofstra experiment. For example, it is impossible to prove that a laboratory experiment is true in 100% of cases. “In the real world, you are limited by the properties of the experimental setup,” Yuyen said. “How to distinguish between the results in 100% and 99.9999%?”

However, leaving aside the practical subtleties, we must admit that Slofstra proved the existence of at least a mathematical method for assessing the fundamental feature of the Universe, which otherwise would remain outside our horizons. When Bell came up with his non-local games, he hoped that they would be useful for sensing one of the most tempting phenomena of the universe. Fifty years later, his invention found even greater depth.