On undertaking the task of writing the book Mathematics and Physics, the author realized that its size would hardly be sufficient to attempt to explain what the “and” means in the title.

Yu. I. Manin, Mathematics and PhysicsFootnote 1

1. Bell’s original inequality and its variations

1.1. Bell’s inequality

If random variables \(\,\xi\), \(\eta\), and \(\,\zeta\) do not exceed unity in absolute value, then the following Bell inequality holds:

$$\mathopen| \operatorname{E} \xi\zeta - \operatorname{E} \eta\zeta| \leq 1 - \operatorname{E} \xi\eta.$$

This statement can be found in [19, Problem 1.4.15]. Here, as is customary in probability theory, the symbol \( \operatorname{E} \) means the expectation of the corresponding random variable.

An elementary purely algebraic inequality that directly implies the probabilistic Bell inequality says that if real numbers \(\xi\), \(\eta\), and \(\zeta\) are such that \(|\xi|\leq 1\), \(|\eta|\leq 1\), and \(|\zeta|\leq 1\), then \(\xi(1+\eta)\leq 1+\eta\); hence \(\xi-\eta\leq 1-\xi\eta\) and \(|\xi\zeta-\eta\zeta|\leq 1-\xi\eta\).

1.2. Clauser–Horne–Shimony–Holt inequality

The following Clauser–Horne–Shimony–Holt inequality [4]Footnote 2 (see also [11]), which results from Bell’s inequality, turned out to be more convenient for physical considerations:

$$\bigl| \operatorname{E} X_1 Y_1 + \operatorname{E} X_1 Y_2 + \operatorname{E} X_2 Y_1 - \operatorname{E} X_2 Y_2\bigr| \leq 2$$

for four random variables whose absolute values do not exceed unity.

1.3. Generalization of Bell’s inequalities

Proposition.

If \(\,|X_k|\leq 1,\) \(k=0,1,\dots,n,\) then for \(\,n \geq 2\) one has

$$|X_0 X_1 - X_{n-1} X_n| \leq (n-1) - \sum_{k=0}^{n-2} X_k X_{k+2}$$

and for \(\,n \geq 3\) one has

$$\Biggl|X_0 X_1 - X_{n-1} X_n + \sum_{k=0}^{n-2}X_k X_{k+2}\Biggr| \leq n-1.$$

Therefore, if \(\,|X_k| \leq 1,\) \(k=0,1,\dots,n,\) then for \(\,n\geq 2\) it always holds that

$$\mathopen| \operatorname{E} X_0 X_1 - \operatorname{E} X_{n-1} X_n| \leq (n-1) - \sum_{k=0}^{n-2} \operatorname{E} X_k X_{k+2}$$

and for \(\,n \geq 3\) it always holds that

$$\Biggl| \operatorname{E} X_0 X_1 - \operatorname{E} X_{n-1} X_n + \sum_{k=0}^{n-2} \operatorname{E} X_k X_{k+2}\Biggr| \leq n-1.$$

Proof.

We prove the above inequalities by step-by-step application of Bell’s inequality with descent from \(n\) to \(1\):

$$\begin{aligned} \, X_0 X_1 - X_{n-1} X_n + \sum_{k=0}^{n-2}X_k X_{k+2} &= X_0 X_1 + \sum_{k=0}^{n-3}X_k X_{k+2} + X_{n-2} X_n - X_{n-1} X_n \\[1pt] &\leq X_0 X_1 + \sum_{k=0}^{n-3}X_k X_{k+2} - X_{n-2} X_{n-1} + 1 \\ &\phantom{={}}{}\dots\dots\dots\dots\dots\dots \\[2pt] &\leq X_0 X_1 - X_0 X_1 + (n-1) = n-1. \end{aligned}$$

By changing the signs of \(X_0,X_3,X_4,X_7,X_8,X_{11},X_{12},\dots\), we change the sign of the left-hand side of the inequality without changing the sign of the right-hand side.

Hence, if \(|X_k| \leq 1\), \(k=0,1,\dots,n\), then for \(n \geq 3\) we have

$$-(n-1) \leq X_0 X_1 - X_{n-1} X_n + \sum_{k=0}^{n-2}X_k X_{k+2} \leq n-1$$

and

$$\Biggl|X_0 X_1 - X_{n-1} X_n + \sum_{k=0}^{n-2}X_k X_{k+2}\Biggr| \leq n-1,$$

as well as

$$\Biggl| \operatorname{E} X_0 X_1 - \operatorname{E} X_{n-1} X_n + \sum_{k=0}^{n-2} \operatorname{E} X_k X_{k+2}\Biggr| \leq n-1.$$

By changing the signs of \(X_0,X_2,X_4,X_6,X_8,\dots\) and using the already proved inequality

$$X_0 X_1 - X_{n-1} X_n \leq (n-1) - \sum_{k=0}^{n-2}X_k X_{k+2},$$

which reduces to Bell’s inequality for \(n=2\), we obtain the relation

$$- (X_0 X_1 - X_{n-1} X_n) \leq (n-1) - \sum_{k=0}^{n-2}X_k X_{k+2}.$$

Therefore, if \(|X_k|\leq 1\), \(k=0,1,\dots,n\), then for \(n\geq 2\) we have

$$|X_0 X_1 - X_{n-1} X_n| \leq (n-1) - \sum_{k=0}^{n-2}X_k X_{k+2}$$

and also

$$\mathopen| \operatorname{E} X_0 X_1 - \operatorname{E} X_{n-1} X_n| \leq (n-1) - \sum_{k=0}^{n-2} \operatorname{E} X_k X_{k+2}.$$

This is a generalization of Bell’s original inequality \(\mathopen| \operatorname{E} \xi\zeta - \operatorname{E} \eta\zeta| \leq 1 - \operatorname{E} \xi\eta\). \(\quad\square\)

In the special case of \(n=3\), the second of the inequalities in the proposition reads

$$\bigl|X_0 X_1 - X_2 X_3 + X_0 X_2 + X_1 X_3\bigr| \leq 2.$$

Introducing here the notation \((X_0,X_1,X_2,X_3) = (X_1,Y_1,Y_2,X_2)\), we obtain

$$\bigl|X_1 Y_1 + X_1 Y_2 + X_2 Y_1 - X_2 Y_2\bigr| \leq 2.$$

This implies the Clauser–Horne–Shimony–Holt inequality

$$\bigl| \operatorname{E} X_1 Y_1 + \operatorname{E} X_1 Y_2 + \operatorname{E} X_2 Y_1 - \operatorname{E} X_2 Y_2\bigr| \leq 2$$

for four random variables whose absolute values do not exceed unity.

2. Source of Bell’s inequality

The elementary inequalities presented above are unlikely to excite any serious interest in themselves; mathematics, and probability theory in particular, knows much more subtle relationships that arouse not only interest but also delight. Bell’s inequality is interesting not so much in itself in its formal mathematical appearance, but because of its origin and the depth of the questions for which it was invented and used.

2.1. Classical determinism or quantum randomness

Physicists, and not only them, are well aware of the almost philosophical dispute between two giants, Einstein and Bohr, which began publicly in 1927. The first argued that “God does not play dice” and the probabilistic predictions of quantum mechanics, strikingly different from the usual determinism of classical physics, are only evidence of the incompleteness of the theory. On the other hand, Bohr argued that the probability here is not at all accidental, that it is a law of nature, and that this is how the quantum world works. It is absolutely remarkable that the protracted and seemingly almost philosophical debate gradually managed to lead to experimentally verifiable mathematical relations. These relations—Bell’s inequalities applied to physical quantities—were written by the theoretical physicist Bell in his 1964 paper [2] devoted to the analysis of the famous Einstein–Podolsky–Rosen (EPR) paradox of 1935 (see [6, 9]), which was supposed to finally convince Bohr and all his followers of the incompleteness of quantum mechanics. Bell’s inequalities for specific experimentally observable quantities were derived under the assumptions of classical physics, and it was shown that if quantum theory is valid, then these inequalities can be violated (Bell’s theorem). Experimental testing of these inequalities immediately began (Clauser and Freedman). It is believed that Alain Aspect’s experiment [1], carried out by him in 1982 in a laboratory near Paris, was completely convincing. In the experiment, Bell’s inequalities were violated, which means that at least one of those “immutable” principles of classical determinism, on which Bell’s inequalities rested, turned out to be incorrect when applied to the microworld.

Let me recall that 2016 marked 100 years since Einstein published his general theory of relativity (1916). But Einstein was also a pioneer of quantum physics,Footnote 3 although, as mentioned above, he radically diverged from Bohr in his general deterministic views. To reinforce them and to demonstrate the incompleteness of the quantum description of reality (nature), the famous EPR thought experiment was invented, which later became the subject of discussion by physicists and mathematicians of several generations. In our time, it has transformed into the idea of quantum communications, quantum information theory, quantum computing, and a quantum computer. Without delving into these already very extensive areas, in which the author is not competent, let me still say at least a few words about the content and conclusions of the EPR experiment.

2.2. Einstein–Podolsky–Rosen paradox, its development, and analysis

One of the cornerstones of quantum theory is the Heisenberg uncertainty principle, which postulates the impossibility of simultaneously accurately measuring certain physical characteristics of a quantum object, such as the position and momentum of a quantum particle. However, in quantum physics, just as in classical physics, the law of conservation of momentum is valid, according to which if some system that had zero momentum breaks up into two identical parts (particles), then these parts, flying apart, have momenta equal in magnitude and opposite in direction. This means that having measured the momentum of one particle, we can find the momentum of the second without measuring it, and we now need to only measure the coordinate of the second particle in order, it would seem, to know both its coordinate and momentum.

Having greatly simplified many things, we still retained the main idea of the thought experiment proposed by Einstein, Podolsky, and Rosen: the idea of using a pair (EPR pair) of particles connected by a common history and some conservation law.

A big simplifying step that made the EPR thought experiment feasible was Bohm’s proposal (1951) to consider not the momentum and coordinate (continuous variables) but rather dichotomous variables, for example, the projection of the spin onto a distinguished direction for a spin \(1/2\) particle.

Quantum systems (particles) for which the measurement has two stable outcomes (for example, the state of the electron spin or the polarization of a photon) are now commonly called qubits (quantum bits, q-bits) by analogy with classical bits of information. It is physical qubits and their combinations that should become information carriers and form the computing base of emerging quantum computers.

Two such quantum particles can form a pair in different ways; more precisely, such a pair can be in different quantum states. In particular, a quantum pair can be, as it is called, entangled. (Physicists call such a pair an EPR pair, remembering the experiment proposed by Einstein, Podolsky, and Rosen.) For example, such a pair can be formed by two electrons with zero total spin. Suppose the electrons fly apart and their spins are measured by a pair of coordinately oriented Stern–Gerlach devices. It turns out that if you have measured the spin of one of the particles of an entangled pair, in our case the spin of one of the electrons (it could initially have any direction), then the spin of the other particle of the pair, when measured by the co-directional Stern–Gerlach device, will automatically turn out to be the opposite.

It is remarkable that, according to quantum theory, the components of such a pair themselves can be far from each other, and the effect will be the same. Then, it would seem, it is possible to transmit information from one place to another at a speed exceeding the speed of light, which contradicts the theory of relativity. Superluminal transfer of information by traditional methods (with the movement of masses, waves, and energy) is impossible, but it is noteworthy that the indicated nonlocal quantum interactions indeed take place! This can be considered a by-product of experiments related to the EPR paradox.

Having considered the EPR experiment in the form proposed by Bohm, Bell did the following. Having accepted the basic principles of classical determinism, he, as we have already said, obtained some inequalities between measured quantities, which should not be violated if the initial assumptions made are valid. On the other hand, he showed that if we proceed from the principles of quantum theory, then these inequalities can be violated. All that remains is to implement the described experiment and check who is right. This was done. It was shown that if we want to correctly describe the world of quantum particles, we will inevitably have to abandon at least one of the principles of classical physics that seemed indisputable and valid on all scales, for example, determinism or the principle of locality of interactions.

2.3. New experiments

Just as the original EPR experiment was simplified by Bohm, Bell’s argument from [2], using the original Bell inequality, was simplified and now based on a Clauser–Horne–Shimony–Holt type inequality similar to the one given above in Subsection 1.2. We will not go into describing the details of the experiments themselves and analyzing their results. One can read about them in [4], and in a modern presentation, for example, first in [18] and then in [11].

For information, we note only the following. The experiments designed to test Bell’s inequalities, for example Aspect’s experiments, were carried out with an EPR pair of entangled quantum particles and were of a statistical nature. Improved experiments continued later as well. In particular, in 1990–2000, Greenberger, Horne, Zeilinger, and Shimony conducted an experiment (GHSZ experiment) with three photons in an entangled state (for a description, see [18, 10]). A remarkable feature of this experiment is that the quantum mechanical predictions of some results of measurements on three entangled particles contradict classical determinism and local realism in those cases where quantum theory gives reliable (i.e., non-statistical) predictions.

Note that Bell’s inequalities were not even used here. The language of complex analysis, which lies at the very foundation of quantum theory with its complex wave function and complex probability amplitude, is at work here, as well as the fact that quantum systems are united according to the tensor law rather than through the direct product of their state spaces.

2.4. Nonlocality of quantum interactions

Note that the nonlocality of quantum interactions (if this is interpreted this way) could seemingly be tested in an experiment simpler than the above-mentioned general experiments designed to test Bell’s inequalities. Suppose there is a source of EPR electron pairs between identically oriented meters (Stern–Gerlach devices) \(A\) and \(B\). The source generates pairs of electrons in an entangled state. One electron of the pair flies towards the meter \(A\), and the other, towards \(B\). The spins of the electrons in each pair are opposite, but in the plane perpendicular to the \(AB\) axis they can have any direction. The distribution of these random directions is uniform, so any of the properly oriented devices \(A\) and \(B\), working alone, will be equally likely to record spin up and down in the direction of the orientation of its vertical axis orthogonal to the direction \(AB\). This results in a typical random sequence of a Bernoulli process. However, if both electrons of a pair are measured, and first one of the devices, say \(A\), does this, then it turns out that the device \(B\) will most definitely register a sequence of spins opposite to the one recorded by \(A\). Moreover, even if the devices \(A\) and \(B\) are installed far from each other, almost at the same distance from the source of EPR pairs, but the device \(B\) is placed a little further so that the device \(A\) is triggered first, then the picture will remain the same, although the possibility of classical transfer of information from \(A\) to \(B\) after measurement in \(A\) but before measurement in \(B\) will be completely excluded (by the theory of relativity).Footnote 4

Let us add that the described quantum phenomenon—if not for superluminal transmission of information, then, for example, for the co-directional orientation of remote devices—can probably already be used. Let us also recall that although it is not possible to control the time-unpredictable act of radioactive decay of a single atom, control of an atomic reactor is quite successfully implemented thanks to the standard (for probability theory) laws of large numbers, when the collective behavior of a large system of random variables turns out to be almost deterministic. This may also apply to the EPR pairs discussed above.

2.5. Final comment (or Digging deeper)

We started with simple mathematical inequalities. Then we explained what physical question had initiated them. We talked about the Einstein–Podolsky–Rosen thought experiment, its development, and field testing in real experiments. The discussion of the issues that arose was far from over. Discussions touch on the idea of physical reality, the foundations of quantum physics, and even the idea of information. A physics-oriented analysis of these issues, as well as an extensive bibliography of the time the article was written, can be found in [12]. I will give just one quote from that paper, which discusses in considerable detail the use of wave function collapse for superluminal information transfer: “Since it [the principle of information transfer] is based on the collapse of wave functions without any motion of matter or wave propagation, the rate of information transfer should not similarly be limited by the velocity of light. However, superluminal information transfer is so unusual, because it affects the main principles of modern physics, that the feasibility (or otherwise) of signal (not wave!) transfer at a speed greater than that of light will be considered in greater detail.”

Modern (and very different) points of view on the deep problems initiated by the EPR paradox are described in [7].

Leaving aside the questions of superluminal information transfer, violation of the principle of causality, and interpretation of the nonlocality of quantum interactions, we can quite definitely state the real successes of the idea of quantum communications, quantum information theory, and quantum computing. Not only many articles but also many books have already been written on this topic. For a first (and quite informative) acquaintance with this range of issues, we can recommend, for example, the (excellently written in our opinion) paper [17]. Among the sources devoted directly to the EPR paradox in the context of Bell’s inequalities, we can mention the reports [22, 16]. These are presentations in the form of slides with short clear statements, bibliography, and even photographs of the main participants in the events. The review [21] may be of interest to mathematicians.

Finally, we note (this is quite remarkable although not widely known) that the very idea of quantum computing and a quantum computer was first expressed not only by the wonderful physicist Richard Feynman [8] but also by the wonderful mathematician Yuri Manin [14].Footnote 5

3. Afterword

This article, apart from this afterword and some additions to the bibliography, was written about five years ago and was posted on my web page http://matan.math.msu.su/vzor/. It is still there today under the same name. During this time, I have become a little more educated, but have not rewritten anything, given that the purely mathematical Section 1, as it seemed to me, did not need any editing (unless one would like to look for a more elegant proof).

However, during this time, two significant events occurred that are directly related to the topic of the article. One of the aces of quantum information theory, Alexander Semenovich Holevo, to whom this volume is dedicated, turned 80 in 2023. And the year before, in 2022, the Nobel Prize in Physics was awarded to Alain Aspect, John Clauser, and Anton Zeilinger for the experiments discussed above to test Bell’s inequalities and for the development and application of ideas dating back to the 1927 discussion between Einstein and Bohr.

Note that Clauser, Aspect, and Zeilinger were awarded the prize for the work they completed approximately 50, 40, and 20 years ago.

Here is the official announcement from the Royal Swedish Academy of Sciences [20]: “The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics 2022 to Alain Aspect, John F. Clauser and Anton Zeilinger ‘for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science’.”

A little less succinctly it looks like this: “Using groundbreaking experiments, Alain Aspect, John Clauser and Anton Zeilinger have demonstrated the potential to investigate and control particles that are in entangled states. What happens to one particle in an entangled pair determines what happens to the other, even if they are really too far apart to affect each other. The laureates’ development of experimental tools has laid the foundation for a new era of quantum technology.”

The last phrase here seems to be the key one. Nobody (not even one of the fathers of quantum physics, Einstein) doubted that quantum mechanics was true. The experiments of the laureates, of course, finally buried the idea of hidden parameters. But it is very important that in the process of improving these experiments, doors to new quantum technologies were opened. They are actively developing.

A desktop quantum computer has not yet been built, but the successes of theorists of quantum computing, quantum information theory, and quantum communication channels, including teleportation, are amazing. Not many years have passed since P. W. Shor’s pioneering work, and now generations of students are already being taught this [11, 13, 17].

And one last thing. So that the reader does not have the impression that the abolition of parameters (variables) supposedly hidden from us completely closes the discussion between Einstein and Bohr, I will quote the opinion of Dirac [5, p. 10]:

“According to the present quantum mechanics, the probability interpretation, the interpretation which was championed by Bohr, is the correct one. But still, Einstein did have a point. He believed that, as he put it, the good God does not play with dice. He believed that basically physics should be of a deterministic character.

And, I think it might turn out that ultimately Einstein will prove to be right, because the present form of quantum mechanics should not be considered as the final form. \(\,\langle\dots\rangle\) It is the best that one can do up till now. But, one should not suppose that it will survive indefinitely into the future. And I think that it is quite likely that at some future time we may get an improved quantum mechanics in which there will be a return to determinism and which will, therefore, justify the Einstein point of view. But such a return to determinism could only be made at the expense of giving up some other basic idea which we now assume without question. We would have to pay for it in some way which we cannot yet guess at, if we are to re-introduce determinism.”

The paper [3], written by the masters of the field, demonstrates the continuation of research into the phenomenon of nonlocality of quantum interactions and quantum information theory.Footnote 6 The paper contains a large bibliography. In particular, the works of A. S. Holevo are cited there.