Simple quantum games reveal the ultimate complexity of the universe.

A game for two can tell if there is an infinite amount of complexity in the universe.

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 tune 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 infinitely?

It is not known to physicists, but in the absence of certainty they make assumptions. And these assumptions differ depending on the field of physics. In one field, physicists assume a finite number of configurations. In the other - the infinite. So far, it is 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 close this question. The games involve two players, isolated from each other. Players ask questions, and win if their answers are in a certain coordinated way. The number of wins is related to the number of different ways of configuring the Universe.
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“There is a philosophical question: of course, or is the number of dimensions of the Universe infinite?” Said Henry Yuyen , a specialist in theoretical computer science from the University of Toronto. "People think that it is impossible to verify this, but one of the possible ways to solve the problem is to use the game William invented."

Yuen speaks of William Slofstra , a mathematician at 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 number of unusual resources — entangled quantum particles — they can always win.

Other researchers have since tweaked the Slofstra result. They proved that in order to achieve the same inference it is not necessary to play a game with hundreds of questions. In 2017, three researchers proved that there are games of just five questions that can be won 100% of the time if the player has access to an unlimited number of entangled particles.

All these games are based on games that were invented more than 50 years ago by the 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 be 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 the question. Players win or lose depending on the compatibility of their answers.

One of these games is the magic square. Players Alice and Bob draw a 3x3 square grid. The judge asks Alice to fill in 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 in one of the columns so that the sum is even. Alice and Bob win if they write the same number at the intersection of their row and column.

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



It seems that in the game with a magic square and other similar games there is no way to win 100% of the time. 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 an electron moving rapidly around a circle. To determine its location, we perform the measurement. But before the measurement, the electron does not have a specific location. It is characterized by a mathematical formula expressing the probability of finding it in a certain 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 a measurement determines the location of one of them at a certain location of a circle, then the other will necessarily be at the opposite point. Such a relationship of two electrons is preserved, and when they are nearby, and when they are separated by many light years. Even at this 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 suggest such a possibility. Albert Einstein ridiculed the intricacies of the famous phrase “frightening long-range action,” and for years argued that this could not be.

To implement a quantum strategy in a magic square game, Alice and Bob take one of the tangled particles. To determine which numbers to write down, they measure the properties of their particle — about the same as if they were throwing related cubes to select answers.


John Stuart Bell, who invented non-local games

Bell calculated, and a number of subsequent experiments showed that using strange quantum particle correlations, players in such games can coordinate their answers much more precisely 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 classical view of the world is incomplete - and at the time such a conclusion was easy to make. “Bell came up with an experiment that can be done in the lab,” said Cleve. If we manage to register a success rate exceeding the expected one, it will become clear that the players use 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, which is played by two players who give 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, let's say, a game for two players, Alice and Bob, who need to match the socks from their dressers. Each player must choose one sock, not knowing which sock the other chooses. Players can not agree in advance on the choice. If their socks are 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 making a pair increase. Based on the choice of color of the sock on the measurement results of one pair of entangled particles, they can coordinate the choice of this one attribute of the sock.

However, they still have to guess about the rest of the attributes - wool is a sock or cotton, as high as the ankle or mid-calf. But using additional entangled 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 sock. As a result, thanks to the ability to coordinate the selection of many attributes, they are more likely to choose socks from one pair.

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

But in the game 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 the amount will give a certain amount.

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 win all the time. But with a pair of entangled particles, the gain would be more constant, as in the socks example.

Slofstre was interested in understanding whether the number of entangled particles exists, beyond which the probability of a team to win ceases to grow. Perhaps players could build an optimal strategy with five pairs of entangled particles, or 500 pairs. “We were hoping that we could say: for optimal play, so much confusion is needed,” Slofstra said. “But it turned out that this is not the case.”

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

Is the universe infinite?

Slofstra's result was a shock to scientists. Eleven days after the appearance of this work, computer scientist Scott Aaronson wrote that the result touches "a question of almost metaphysical importance: namely, what experiments can in principle show whether the Universe is discrete or continuous?"

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


William Slofstra, a mathematician from 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, the state space is even in the box with socks. Each sock can be described in 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 no matter how large 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 that are independent of each other, which can be observed in principle,” said Tomas Vidic , an IT specialist from 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 designate the probability of each point of the circle. Since there are an infinite number of points, the state space describing the location of an electron will have an infinite number of dimensions.

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

But perhaps the idea of ​​an infinite space of dimensions 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, Beckenstein's limit , says that if a black hole does not need 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 nature should be pixelated on the smallest scale. But if electrons require space of states with an infinite number of dimensions, physical reality is inherently continuous even at the smallest resolution.

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

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

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

However, leaving aside the practical subtleties, it must be admitted that Slofstra proved the existence of at least a mathematical way of assessing the fundamental peculiarities of the Universe, which otherwise would have remained outside of our horizons. When Bell invented his non-local games, he hoped that they would be useful for probing one of the most enticing phenomena of the universe. Fifty years later, his invention found even greater depth.