Particle entanglement. What is quantum entanglement? The essence in simple words. Is teleportation possible? Quantum entanglement theory

• Translation

Quantum entanglement is one of the most complex concepts in science, but its basic principles are simple. And if understood, entanglement opens the way to a better understanding of concepts such as the plurality of worlds in quantum theory.

An enchanting aura of mystery surrounds the concept of quantum entanglement, as well as (somehow) the related requirement of quantum theory that there must be “many worlds.” And yet, at their core, these are scientific ideas with down-to-earth meaning and specific applications. I would like to explain the concepts of entanglement and many worlds as simply and clearly as I know them.

I

Entanglement is considered a phenomenon unique to quantum mechanics– but that’s not true. In fact, it may be more understandable to begin with (although this is an unusual approach) to consider a simple, non-quantum (classical) version of entanglement. This will allow us to separate the subtleties associated with entanglement itself from other oddities of quantum theory.

Entanglement occurs in situations in which we have partial information about the state of two systems. For example, two objects can become our systems – let’s call them kaons. "K" will stand for "classical" objects. But if you really want to imagine something concrete and pleasant, imagine that these are cakes.

Our kaons will have two shapes, square or round, and these shapes will indicate their possible states. Then the four possible joint states of the two kaons will be: (square, square), (square, circle), (circle, square), (circle, circle). The table shows the probability of the system being in one of the four listed states.

We will say that kaons are “independent” if knowledge about the state of one of them does not give us information about the state of the other. And this table has such a property. If the first kaon (cake) is square, we still don't know the shape of the second one. Conversely, the form of the second tells us nothing about the form of the first.

On the other hand, we will say that two kaons are entangled if information about one of them improves our knowledge about the other. The second tablet will show us strong confusion. In this case, if the first kaon is round, we will know that the second one is also round. And if the first kaon is square, then the second one will be the same. Knowing the shape of one, we can unambiguously determine the shape of the other.

The quantum version of entanglement looks essentially the same - it is a lack of independence. In quantum theory, states are described by mathematical objects called wave functions. The rules that combine wave functions with physical possibilities give rise to very interesting complications that we will discuss later, but the basic concept of entangled knowledge that we demonstrated for the classical case remains the same.

Although brownies cannot be considered quantum systems, entanglement in quantum systems occurs naturally, such as after particle collisions. In practice, unentangled (independent) states can be considered rare exceptions, since correlations arise between them when systems interact.

Consider, for example, molecules. They consist of subsystems - specifically, electrons and nuclei. The minimum energy state of a molecule, in which it usually exists, is a highly entangled state of electrons and nucleus, since the arrangement of these constituent particles will not be independent in any way. When the nucleus moves, the electron moves with it.

Let's return to our example. If we write Φ■, Φ● as wave functions describing system 1 in its square or round states and ψ■, ψ● for wave functions describing system 2 in its square or round states, then in our working example all states can be described , How:

Independent: Φ■ ψ■ + Φ■ ψ● + Φ● ψ■ + Φ● ψ●

Entangled: Φ■ ψ■ + Φ● ψ●

The independent version can also be written as:

(Φ■ + Φ●)(ψ■ + ψ●)

Note how in the latter case the brackets clearly separate the first and second systems into independent parts.

There are many ways to create entangled states. One is to measure a composite system that gives you partial information. One can learn, for example, that two systems have agreed to be of the same form without knowing which form they have chosen. This concept will become important a little later.

The more common effects of quantum entanglement, such as the Einstein-Podolsky-Rosen (EPR) and Greenberg-Horn-Seilinger (GHZ) effects, arise from its interaction with another property of quantum theory called the complementarity principle. To discuss EPR and GHZ, let me first introduce this principle to you.

Up to this point, we have imagined that kaons come in two shapes (square and round). Now let’s imagine that they also come in two colors – red and blue. Considering classical systems, for example, cakes are additional property would mean that the kaon can exist in one of four possible states: red square, red circle, blue square and blue circle.

But quantum cakes are quantons... Or quantons... They behave completely differently. The fact that a quanton in some situations may have different shapes and color does not necessarily mean that it has both form and color at the same time. Actually, common sense, which Einstein demanded from physical reality, does not correspond to experimental facts, as we will soon see.

We can measure the shape of a quanton, but in doing so we will lose all information about its color. Or we can measure the color, but lose information about its shape. According to quantum theory, we cannot measure both shape and color at the same time. No one's view of quantum reality is complete; we have to take into account many different and mutually exclusive pictures, each of which has its own incomplete picture of what is happening. This is the essence of the principle of complementarity, as formulated by Niels Bohr.

As a result, quantum theory forces us to be careful in attributing properties to physical reality. To avoid contradictions, we must admit that:

A property does not exist unless it is measured.
Measurement – active process, changing the measured system

II

Now we will describe two exemplary, but not classical, illustrations of the oddities of quantum theory. Both have been tested in rigorous experiments (in real experiments people do not measure the shapes and colors of cakes, but the angular momenta of electrons).

Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) described a surprising effect that occurs when two quantum systems become entangled. The EPR effect combines a special, experimentally achievable form of quantum entanglement with the principle of complementarity.

An EPR pair consists of two quantons, each of which can be measured in shape or color (but not both at once). Suppose we have many such pairs, all of them the same, and we can choose what measurements we make on their components. If we measure the shape of one member of an EPR pair, we are equally likely to get a square or a circle. If we measure color, we are equally likely to get red or blue.

Interesting effects that seemed paradoxical to EPR arise when we measure both members of the pair. When we measure the color of both members, or their shape, we find that the results are always the same. That is, if we discover that one of them is red and then measure the color of the second, we also discover that it is red - and so on. On the other hand, if we measure the shape of one and the color of the other, no correlation is observed. That is, if the first one was a square, then the second one could be blue or red with equal probability.

According to quantum theory, we will obtain such results even if the two systems are separated by a huge distance and the measurements are carried out almost simultaneously. The choice of measurement type at one location appears to affect the state of the system at another location. This “frightening action at a distance,” as Einstein called it, apparently requires the transmission of information—in our case, information about a measurement being made—faster than the speed of light.

But is it? Until I know what results you got, I don't know what to expect. I get useful information when I find out your result, not when you take the measurement. And any message containing the result you received must be transmitted in some way physically, slower than the speed of light.

With further study, the paradox collapses even more. Let's consider the state of the second system if the measurement of the first gave a red color. If we decide to measure the color of the second quanton, we will get red. But by the principle of complementarity, if we decide to measure its shape when it is in the "red" state, we have an equal chance of getting a square or a circle. Therefore, the result of EPR is logically predetermined. This is simply a restatement of the principle of complementarity.

There is no paradox in the fact that distant events are correlated. After all, if we put one of two gloves from a pair into boxes and send them to different ends of the planet, it is not surprising that by looking in one box, I can determine which hand the other glove is intended for. Likewise, in all cases, the correlation of EPR pairs must be recorded on them when they are nearby so that they can withstand subsequent separation, as if having memory. The strangeness of the EPR paradox is not in the possibility of correlation itself, but in the possibility of its preservation in the form of additions.

III

Daniel Greenberger, Michael Horn and Anton Zeilinger discovered another beautiful example of quantum entanglement. IT includes three of our quantons, which are in a specially prepared entangled state (GHZ-state). We distribute each of them to different remote experimenters. Each of them chooses, independently and randomly, whether to measure color or shape and records the result. The experiment is repeated many times, but always with three quantons in the GHZ state.

Each individual experimenter obtains random results. Measuring the shape of a quanton, he obtains with equal probability a square or a circle; when measuring the color of a quanton, it is equally likely to be red or blue. So far everything is ordinary.

But when experimenters get together and compare the results, the analysis shows a surprising result. Let's say we call square shape and red color is “good”, and circles and Blue colour- “evil.” Experimenters find that if two of them decide to measure shape and the third decides to measure color, then either 0 or 2 of the measurements are “evil” (i.e., round or blue). But if all three decide to measure a color, then either 1 or 3 dimensions are evil. This is what quantum mechanics predicts, and this is exactly what happens.

Question: Is the amount of evil even or odd? Both possibilities are realized in different dimensions. We have to abandon this issue. It makes no sense to talk about the amount of evil in a system without relating it to how it is measured. And this leads to contradictions.

The GHZ effect, as physicist Sidney Coleman describes it, is “a slap in the face from quantum mechanics.” It breaks down the conventional, experiential expectation that physical systems have predetermined properties independent of their measurement. If this were so, then the balance of good and evil would not depend on the choice of measurement types. Once you accept the existence of the GHZ effect, you will not forget it, and your horizons will be expanded.

IV

For now, we are discussing how entanglement prevents us from assigning unique independent states to multiple quantons. The same reasoning applies to changes in one quanton that occur over time.

We talk about “entangled histories” when it is impossible for a system to be assigned a certain state at each moment in time. Just as in traditional entanglement we rule out possibilities, we can create entangled histories by making measurements that collect partial information about past events. In the simplest entangled stories we have one quanton that we study at two different points in time. We can imagine a situation where we determine that the shape of our quanton was square both times, or round both times, but both situations remain possible. This is a temporal quantum analogy to the simplest versions of entanglement described earlier.

Using a more complex protocol, we can add a little extra detail to this system, and describe situations that trigger the "many-worlds" property of quantum theory. Our quanton can be prepared in the red state, and then measured and obtained in blue. And as in the previous examples, we cannot permanently assign a quanton the property of color in the interval between two dimensions; he doesn't have either a certain form. Such stories are realized, limited, but completely controlled and in an exact way, the intuition inherent in the picture of the multiplicity of worlds in quantum mechanics. A certain state can be divided into two contradictory historical trajectories, which then connect again.

Erwin Schrödinger, the founder of quantum theory, who was skeptical about its correctness, emphasized that the evolution of quantum systems naturally leads to states, the measurement of which can provide extremely different results. His thought experiment with "Schrodinger's cat" famously postulates quantum uncertainty, brought to the level of influence on feline mortality. Before measuring, it is impossible to assign the property of life (or death) to a cat. Both, or neither, exist together in an otherworldly world of possibility.

Everyday language is ill-suited to explain quantum complementarity, in part because everyday experience does not include it. Practical cats interact with surrounding air molecules, and other objects, in completely different ways, depending on whether they are alive or dead, so in practice the measurement takes place automatically, and the cat continues to live (or not live). But the stories describe the quantons, which are Schrödinger's kittens, with confusion. Their Full description requires that we consider two mutually exclusive trajectories of properties.

Controlled experimental implementation of entangled stories is a delicate thing, since it requires the collection of partial information about quantons. Conventional quantum measurements typically collect all the information at once—determining an exact shape or a precise color, for example—rather than obtaining partial information several times. But it can be done, albeit with extreme technical difficulties. In this way we can assign a certain mathematical and experimental meaning to the extension of the concept of “many worlds” in quantum theory, and demonstrate its reality.

Hello, dear readers! Welcome to the blog!

What is quantum entanglement in simple words? Teleportation - is it possible? Has the possibility of teleportation been experimentally proven? What is Einstein's nightmare? In this article you will get answers to these questions.

Introduction

We often encounter teleportation in science fiction films and books. Have you ever wondered why what writers came up with eventually becomes our reality? How do they manage to predict the future? I think this is not an accident. Science fiction writers often have extensive knowledge of physics and other sciences, which, combined with their intuition and extraordinary imagination, helps them construct a retrospective analysis of the past and simulate future events.

From the article you will learn:

• What is quantum entanglement?
• Einstein's dispute with Bohr. Who is right?
• Is teleportation confirmed experimentally?

Concept "quantum entanglement" arose from a theoretical assumption arising from the equations of quantum mechanics. It means this: if 2 quantum particles (they can be electrons, photons) turn out to be interdependent (entangled), then the connection remains, even if they are separated into different parts of the Universe

The discovery of quantum entanglement goes some way to explaining the theoretical possibility of teleportation.

In short, then spin of a quantum particle (electron, photon) is called its own angular momentum. Spin can be represented as a vector, and the quantum particle itself as a microscopic magnet.

It is important to understand that when no one observes a quantum, for example an electron, then it has all the spin values ​​at the same time. This fundamental concept of quantum mechanics is called “superposition.”

Imagine that your electron is spinning clockwise and counterclockwise at the same time. That is, he is in both states of spin at once (vector spin up/vector spin down). Introduced? OK. But as soon as an observer appears and measures its state, the electron itself determines which spin vector it should accept - up or down.

Want to know how electron spin is measured? It is placed in a magnetic field: electrons with spin opposite the direction of the field, and with spin in the direction of the field, will be deflected in different sides. Photon spins are measured by directing them into a polarizing filter. If the spin (or polarization) of the photon is “-1”, then it does not pass through the filter, and if it is “+1”, then it does.

Summary. Once you have measured the state of one electron and determined that its spin is “+1”, then the electron associated or “entangled” with it takes on a spin value of “-1”. And instantly, even if he is on Mars. Although before measuring the state of the 2nd electron, it had both spin values ​​simultaneously (“+1” and “-1”).

This paradox, proven mathematically, did not like Einstein very much. Because it contradicted his discovery that there is no speed greater than the speed of light. But the concept of entangled particles proved: if one of the entangled particles is on Earth, and the 2nd is on Mars, then the 1st particle at the moment its state is measured instantly ( faster speed light) transmits information to the 2nd particle about what spin value it should take. Namely: the opposite meaning.

Einstein's dispute with Bohr. Who is right?

Einstein called “quantum entanglement” SPUCKHAFTE FERWIRKLUNG (German) or frightening, ghostly, supernatural action at a distance.

Einstein did not agree with Bohr's interpretation of quantum particle entanglement. Because it contradicted his theory that information cannot be transmitted faster than the speed of light. In 1935 he published an article describing thought experiment. This experiment was called the “Einstein-Podolsky-Rosen Paradox.”

Einstein agreed that bound particles could exist, but came up with a different explanation for the instantaneous transfer of information between them. He said "entangled particles" rather like a pair of gloves. Imagine that you have a pair of gloves. You put the left one in one suitcase, and the right one in the second. You sent the 1st suitcase to a friend, and the 2nd to the Moon. When the friend receives the suitcase, he will know that the suitcase contains either a left or right glove. When he opens the suitcase and sees that there is a left glove in it, he will instantly know that there is a right glove on the Moon. And this does not mean that the friend influenced the fact that the left glove is in the suitcase and does not mean that the left glove instantly transmitted information to the right one. This only means that the properties of the gloves were originally the same from the moment they were separated. Those. entangled quantum particles initially contain information about their states.

So who was Bohr right when he believed that bound particles transmit information to each other instantly, even if they are separated over vast distances? Or Einstein, who believed that there is no supernatural connection, and everything is predetermined long before the moment of measurement.

This debate moved into the field of philosophy for 30 years. Has the dispute been resolved since then?

Bell's theorem. Is the dispute resolved?

John Clauser, while still a graduate student at Columbia University, found forgotten work Irish physicist John Bell. It was a sensation: it turns out Bell managed to break the deadlock between Bohr and Einstein.. He proposed experimentally testing both hypotheses. To do this, he proposed building a machine that would create and compare many pairs of entangled particles. John Clauser began to develop such a machine. His machine could create thousands of pairs of entangled particles and compare them according to different parameters. The experimental results proved Bohr was right.

And soon the French physicist Alain Aspe conducted experiments, one of which concerned the very essence of the dispute between Einstein and Bohr. In this experiment, the measurement of one particle could directly affect another only if the signal from the 1st to the 2nd passed at a speed exceeding the speed of light. But Einstein himself proved that this is impossible. There was only one explanation left - an inexplicable, supernatural connection between the particles.

The experimental results proved that the theoretical assumption of quantum mechanics is correct. Quantum entanglement is a reality ( Quantum entanglement Wikipedia). Quantum particles can be connected despite vast distances. Measuring the state of one particle affects the state of the 2nd particle located far from it as if the distance between them did not exist. Supernatural long-distance communication actually happens.

The question remains, is teleportation possible?

Is teleportation confirmed experimentally?

Back in 2011, Japanese scientists were the first in the world to teleport photons! A beam of light was instantly moved from point A to point B.

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Quantum entanglement, or “spooky action at a distance” as Albert Einstein called it, is a quantum mechanical phenomenon in which the quantum states of two or more objects are interdependent. This dependence persists even if the objects are many kilometers away from each other. For example, you can entangle a pair of photons, take one of them to another galaxy, and then measure the spin of the second photon - and it will be opposite to the spin of the first photon, and vice versa. They are trying to adapt quantum entanglement for instant transmission of data over gigantic distances or even for teleportation.

Physicists from the Scottish University of Glasgow reported an experiment in which scientists were able to obtain the first ever photograph of particles. A phenomenon so strange by physics standards that even a great 20th century scientist nicknamed it “spooky action at a distance.” The achievement of Scottish scientists is very important for the development of new technologies. Why? Let's figure it out.

We have already written more than once that devices are being tested every now and then in different parts of the world. quantum communication. It would seem that all this will not go beyond experiments soon, but, as the Xinhua news agency reports, China has completed the creation of the country's first commercial ultra-secure quantum communication network. Commissioning is planned in the very near future.

There are many popular articles, which talks about quantum entanglement. Experiments with quantum entanglement are very impressive, but have not received any prizes. Why are such experiments interesting for the average person not of interest to scientists? Popular articles talk about amazing properties pairs of entangled particles - impact on one leads to an instant change in the state of the second. And what is hidden behind the term “quantum teleportation”, which has already begun to be said that it occurs with superluminal speed. Let's look at all this from the point of view of normal quantum mechanics.

What comes from quantum mechanics

Quantum particles can be in two types of states, according to classic textbook Landau and Lifshitz - pure and mixed. If a particle does not interact with other quantum particles, it is described by a wave function that depends only on its coordinates or momenta - this state is called pure. In this case, the wave function obeys the Schrödinger equation. Another option is possible - the particle interacts with other quantum particles. In this case, the wave function refers to the entire system of interacting particles and depends on all their dynamic variables. If we are interested in only one particle, then its state, as Landau showed 90 years ago, can be described by a matrix or density operator. The density matrix obeys an equation similar to the Schrödinger equation

Where is the density matrix, H is the Hamiltonian operator, and the brackets denote the commutator.

Landau brought him out. Any physical quantities related to a given particle can be expressed through the density matrix. This condition is called mixed. If we have a system of interacting particles, then each of the particles is in a mixed state. If the particles scatter over long distances and the interaction disappears, their state will still remain mixed. If each of several particles is in a pure state, then the wave function of such a system is the product of the wave functions of each of the particles (if the particles are different. For identical particles, bosons or fermions, it is necessary to make a symmetric or antisymmetric combination, see, but more on that later. The identity of particles, fermions and bosons is already a relativistic quantum theory.

An entangled state of a pair of particles is a state in which there is a constant correlation between physical quantities belonging to different particles. A simple and most common example is that a certain total physical quantity is conserved, for example, the total spin or the angular momentum of a pair. In this case, a pair of particles is in a pure state, but each of the particles is in a mixed state. It may seem that a change in the state of one particle will immediately affect the state of another particle. Even if they are scattered far away and do not interact, this is what is expressed in popular articles. This phenomenon has already been dubbed quantum teleportation. Some illiterate journalists even claim that the change occurs instantly, that is, it spreads faster than the speed of light.

Let's consider this from the point of view of quantum mechanics. Firstly, any impact or measurement that changes the spin or angular momentum of only one particle immediately violates the law of conservation of the total characteristic. The corresponding operator cannot commute with full spin or full angular momentum. Thus, the initial entanglement of the state of a pair of particles is disrupted. The spin or momentum of the second particle can no longer be unambiguously associated with that of the first. We can look at this problem from another angle. After the interaction between particles has disappeared, the evolution of the density matrix of each particle is described by its own equation, in which the dynamic variables of the other particle are not included. Therefore, the impact on one particle will not change the density matrix of the other.

There is even Eberhard's theorem, which states that the mutual influence of two particles cannot be detected by measurements. Let there be a quantum system that is described by a density matrix. And let this system consist of two subsystems A and B. Eberhard’s theorem states that no measurement of observables associated only with subsystem A does not affect the result of measurement of any observables that are associated only with subsystem B. However, the proof of the theorem uses the reduction hypothesis wave function, which has not been proven either theoretically or experimentally. But all these arguments were made within the framework of non-relativistic quantum mechanics and relate to different, non-identical particles.

These arguments don't work in relativistic theory in the case of a pair of identical particles. Let me remind you once again that the identity or indistinguishability of particles comes from relativistic quantum mechanics, where the number of particles is not conserved. However, for slow particles we can use the simpler apparatus of nonrelativistic quantum mechanics, simply by allowing for the indistinguishability of the particles. Then the wave function of the pair must be symmetric (for bosons) or antisymmetric (for fermions) with respect to the permutation of particles. Such a requirement arises in relativistic theory, regardless of particle velocities. It is this requirement that leads to long-range correlations between pairs of identical particles. In principle, a proton and an electron can also be in an entangled state. However, if they diverge by several tens of angstroms, then interaction with electromagnetic fields and other particles will destroy this state. Exchange interaction (as this phenomenon is called) acts at macroscopic distances, as experiments show. A pair of particles, even having separated by meters, remains indistinguishable. If you make a measurement, then you do not know exactly which particle the measured value belongs to. You are taking measurements on a couple of particles at the same time. Therefore, all spectacular experiments were carried out with exactly the same particles - electrons and photons. Strictly speaking, this is not exactly the entangled state that is considered within the framework of non-relativistic quantum mechanics, but something similar.

Let's consider the simplest case - a pair of identical non-interacting particles. If the velocities are small, we can use nonrelativistic quantum mechanics, taking into account the symmetry of the wave function with respect to the permutation of particles. Let the wave function of the first particle , the second particle - , where and are the dynamic variables of the first and second particles, in the simplest case - just coordinates. Then the wave function of the pair

The + and – signs refer to bosons and fermions. Let's assume that the particles are far away from each other. Then they are localized in distant regions 1 and 2, respectively, that is, outside these regions they are small. Let's try to calculate the average value of some variable of the first particle, for example, coordinates. For simplicity, we can imagine that the wave functions include only coordinates. It turns out that the average value of the coordinates of particle 1 lies BETWEEN regions 1 and 2, and it coincides with the average value for particle 2. This is actually natural - the particles are indistinguishable, we cannot know which particle has the coordinates measured. In general, all average values ​​for particles 1 and 2 will be the same. This means that by moving the localization region of particle 1 (for example, the particle is localized inside a defect in the crystal lattice, and we move the entire crystal), we influence particle 2, although the particles do not interact in the usual sense - through an electromagnetic field, for example. This is a simple example of relativistic entanglement.

There is no instantaneous transfer of information due to these correlations between the two particles. The apparatus of relativistic quantum theory was initially constructed in such a way that events located in space-time on opposite sides of the light cone cannot influence each other. Simply put, no signal, no influence or disturbance can travel faster than light. Both particles are actually states of the same field, for example, electron-positron. By influencing the field at one point (particle 1), we create a disturbance that propagates like waves on water. In non-relativistic quantum mechanics, the speed of light is considered infinitely large, which gives rise to the illusion of instantaneous change.

The situation when particles separated by large distances remain bound in pairs seems paradoxical due to classical ideas about particles. We must remember that it is not particles that really exist, but fields. What we think of as particles are simply states of these fields. The classical idea of ​​particles is completely unsuitable in the microworld. Questions immediately arise about size, shape, material and structure elementary particles. In fact, situations that are paradoxical for classical thinking also arise with one particle. For example, in the Stern-Gerlach experiment, a hydrogen atom flies through a non-uniform magnetic field directed perpendicular to the speed. The nuclear spin can be neglected due to the smallness of the nuclear magneton, even if the electron spin is initially directed along the velocity.

The evolution of the wave function of an atom is not difficult to calculate. The initial localized wave packet splits into two identical ones, flying symmetrically at an angle to the original direction. That is, an atom, a heavy particle, usually considered as classical with a classical trajectory, split into two wave packets that can fly apart over quite macroscopic distances. At the same time, I will note that from the calculation it follows that even the ideal Stern-Gerlach experiment is not able to measure the spin of a particle.

If the detector binds a hydrogen atom, for example, chemically, then the “halves” - two scattered wave packets - are collected into one. How does such localization of a smeared particle occur - separately existing theory, which I don't understand. Those interested can find extensive literature on this issue.

Conclusion

The question arises: what is the meaning of numerous experiments demonstrating correlations between particles at large distances? In addition to confirming quantum mechanics, which no normal physicist has doubted for a long time, this is a spectacular demonstration that impresses the public and amateur officials who allocate funds for science (for example, the development of quantum communication lines is sponsored by Gazprombank). For physics, these expensive demonstrations do not yield anything, although they allow the development of experimental techniques.

Literature
1. Landau, L. D., Lifshits, E. M. Quantum mechanics (non-relativistic theory). - 3rd edition, revised and expanded. - M.: Nauka, 1974. - 752 p. - (“ Theoretical physics", volume III).
2. Eberhard, P.H., “Bell’s theorem and the different concepts of nonlocality,” Nuovo Cimento 46B, 392-419 (1978)