It all began at the end of the 19th century, when the scientistfrom France, Henri Poincaré, studied various parts of systems that can be fully analyzed. As usual, this does not sound so difficult, but it was his works that formed the basis of a great task and became one of the mysteries, which scientists of our time call the "Millennium Problems". I think you can easily agree that if you wait enough time, the planets in the sky will line up in the line you need. It will be the same with gas or liquid particles, which can change their position as much as they like, but theoretically at one of the moments in time they will line up relative to each other as they were at the moment of the beginning of measurements. In words, everything is simple - sooner or later it will happen, otherwise it cannot be. But it is rather difficult to prove it in practice. This is what Henri Poincaré worked on more than a century ago. Later, his theories were proven, but that did not make them less interesting.

Who is Henri Poincaré

**Jules Henri Poincaré** (fr.Jules Henri Poincaré was born on April 29, 1854 in Nancy, France and died on July 17, 1912 in Paris, France. He was a French scientist with interests in a wide variety of sciences. Among them were: mathematics, mechanics, physics, astronomy and philosophy.

In addition to his research, HenriPoincaré in different years was also the head of the Paris Academy of Sciences, a member of the French Academy and more than 30 other world academies, including a foreign corresponding member of the St. Petersburg Academy of Sciences.

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Almost unanimously, historians call HenriPoincaré is one of the greatest mathematicians of all time. He was put on a par with Hilbert, the last universal mathematician, a scientist capable of covering all the mathematical results of his time.

Peru Henri Poincaré owns over 500 articles andbooks. All this speaks of him as a genius who, even more than 100 years after his death, can change the world of the future with his theories, formulas, reasoning and other scientific works.

What is Poincaré's return theorem

**Poincaré's return theorem** - one of the basic theories of ergodic theory.Its essence is that under a measure-preserving mapping of space onto itself, almost every point will return to its initial neighborhood. This will take a huge, but finite amount of time.

On the one hand, everything is logical, but thistheory and a little incomprehensible consequence. For example, we have a vessel that is divided into two compartments by a partition. One contains gas, and the other contains nothing. If you remove the baffle, the gas will fill the entire vessel. According to the theory of repetition, sooner or later **all gas particles must line up in the original sequence** in half of the vessel.

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A little unties the hands that time thatit will take, maybe very large. But such a consequence is not entirely correct, since the observation conditions have changed. But if we say that we will not remove the partition, the volume of the gas will not change and it will not have to violate the laws of physics, arbitrarily changing its density, and sooner or later the gas particles will indeed occupy the places in which they were at the start of observations ...

Poincaré theory in a quantum system

If we are talking about the fact that in the traditional systemrepetitions are possible and even inevitable, then we can assume that in a quantum system in which several states are possible, everything is slightly different. It turns out that it is not, and **Poincaré's works can be applied to quantum systems**... However, the rules will be slightly different.

The problem with the application is thatthe state of a quantum system, which consists of a large number of particles, cannot be measured with great precision, let alone perfect measurement. Moreover, we can say that particles in such systems can be considered as completely independent objects. Given the confusion, it is not difficult to understand that there are many complexities to be faced when analyzing such systems.

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Despite this, scientists would not be scientists ifwould not try to demonstrate the effect of Poincaré repetition, including in quantum systems. They did it. But so far this is only possible for systems with a very small number of particles. Their condition must be measured as accurately as possible and must be taken into account.

Say that **it's hard to do** - say nothing.The main difficulty is that the time it takes for the system to return to its original state will increase dramatically even with a slight increase in the number of particles. That is why some scientists do not analyze the system as a whole, but its individual particles. They are trying to understand whether it is possible to return to the original meaning of some parts of this system.

To do this, they study and analyze behaviorultracold gas. It is made up of thousands of atoms and is held in place by electromagnetic fields. The characteristics of such a quantum gas can be described in several quantities. They talk about how closely particles can be bound by the effects of quantum mechanics. In ordinary life, this is not so important and may even seem like something unnecessary, but in quantum mechanics it is crucial.

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As a result, if you understand how such quantitiescharacterize the system as a whole, it will be possible to talk about the possibility of a quantum return. Having received such knowledge, we can more safely say that we know what a gas is, what processes occur in it, and even predict the consequences of exposure to it.

Recently, scientists have been able to prove that **quantum states can return**but some amendments to the concept of repetitionit's still worth making. You should not try to measure the entire quantum system as a whole, because this task is close to impossible. It would be much more correct to focus on some of its elements that can be measured and predicted the behavior of the system as a whole.

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More boldly, such studies anddevelopments in the field of various sciences bring the creation of a real quantum computer closer, and not the test systems that exist now. If the matter moves forward, then a great future awaits us. And at first it seemed that it was just a measurement of something incomprehensible. Is not it?