1.1 Background

Three physicists, Alain Aspect, John F. Clauser, and Anton Zeilinger, are awarded the Nobel Prize in physics in 2022 for “Experiments with entangled photons, establishing the violation of Bell Inequalities and pioneering quantum information science.” This will undoubtedly add impetus to the development of new quantum technology. Anders lrback, Chairman of the Nobel Committee for Physics, pointed out that “The laureates’ work with entangled states is of great importance, even beyond the fundamental questions about the interpretation of quantum mechanics.” It can be understood that the basic mechanism and principles of quantum computing and quantum information processing are based directly on this year’s Nobel Prize work.

In recent years, quantum technology has become one of the commanding heights of scientific and technological competition. A lot of countries around the world have invested huge resources in quantum computing, quantum metrology, quantum communication, and other quantum technological areas. On the other hand, quantum mechanics has always been of interests for public. People usually use quantum mechanics concepts such as “quantum entanglement” and “Schrödinger’s cat” to describe some daily phenomena in Internet. So, what is the meaning of quantum entanglement, and why did the experiments on Bell’s inequality win Nobel Prize in physics? We try to provide some interpretations of these results.

1.2 Quantum entanglement and EPR paradox

It all starts with the mysterious quantum entanglement. In 1935, Schrödinger found that under the framework of quantum mechanics, there would be such a quantum state that the wave functions of two particles could not be written as the direct product of the wave functions of single particles. Namely, the wave function of two particles cannot be separated, and we could only use the whole wave function to describe the two-particle state. This property seems not to be strange. However, when taking account of the quantum mechanics statements about measurement and spatial locality, it would lead to the EPR (Albert Einstein, Boris Podolsky, and Nathan Rosen) paradox [1] that confuses a lot of people. Consider that we can prepare a special entangled state, in which each particle can be measured to two states: “ + ” and “ − .” According to theory that wave function collapses, in such an EPR state, when we find the first particle as “ + ” in measurement, the second particle must be measured as “ − ” and vice versa. Note, however, that the collapse of the wave function in quantum mechanics is instantaneous. Even if we separate the two particles sufficiently far away, the measurement of one particle can also immediately determine the result of other. It is as if there exists an interaction with speed exceeding the speed of light (action at a distance). This result has been absurd to most people, even Einstein and the co-authors, hence named “EPR paradox.” Einstein believed that the world is local or realistic or both, where “local” means that no influence can be generated beyond the speed of light, while reality describes the objective existence of a physical element, and our measurements are all derived from this physical reality and cannot affect it. But in quantum mechanics, when we make a measurement, the wave function will collapse immediately, and the single-shot measurement result is random, even still constrained by the corresponding amplitudes. To illustrate this issue, Einstein et al. argued that this result of quantum mechanics is due to that its theory is incomplete.

This incompleteness is manifested in the hidden variable theory proposed by Boehm in 1952 [2]. In this theory, quantum mechanics is still local-realistic, and the EPR paradox is caused by our ignorance of hidden variables. To understand hidden variables, let us take a daily example. Suppose Alice has a pair of shoes, and she will give them to two people who are far apart from each other, Bob and Charlie. Alice has told them there is only one pair of shoes. Now, when Bob opens his package, he can determine immediately whether Charlie’s shoe is left or right, which seems to be much like quantum entanglement. However, on second thoughts, it is clear that this is not an action at a distance; it is just because Bob knows the implicit condition: there is only one pair of shoes. It is that implicit condition that makes relevant of Bob and Charlie’s results. Quantum mechanics in Boehm’s theory is more like that the world is still local-realistic, but the condition such as “one pair of shoes” is not known; perhaps in the EPR case, the separation of the two particles has already been determined to be opposite by hidden variables, rather than by the action at a distance in measuring. The question then becomes how can we know whether quantum mechanics is described by hidden variables? That is what Bell’s inequalities solve.

1.3 Bell’s inequalities

The significance of Bell’s inequality is to transfer the problem of whether the world is local-realistic into a mathematical formula. And, we can test it by designing specialized physical experiments. The key to solve the problem is that the correlation generated by local hidden variable theory has a bound. If this bound is violated, the world is not described by the local hidden variables; quantum mechanics is complete but nonlocal. The first Bell inequality was proposed by Bell in 1964 [3], and many similar inequalities have been developed in similar framework. The Nobel Prize of physics in this year was awarded for a series of experiments, such as those verified the violation of Clauser, Home, Shimony, and Holt (CHSH) inequalities [4].

CHSH inequality, consider that Alice and Bob are separated by a certain distance, and each has a particle. The state of these two particles may be described by quantum mechanics or by hidden variable theory. Suppose Alice measures the particle in two ways. Let us label it with\(x\), and its value can be\({x}_{0} \mathrm{and} {x}_{1}\). Bob also has two ways of measuring y, labeled as\({y}_{0} \mathrm{and }{y}_{1}\). Moreover, it is assumed that the measurement can only obtain two outcomes: − 1 and + 1. Here, we use \(a\) to represent the measured results of Alice and \(b\) for that of Bob. Then after many measurements, we can find two facts: (1) when the measurement basis is fixed, the results can present a certain probability distribution. For example, when Alice measures \({x}_{0}\) and Bob measures\({y}_{0}\), the values of \(a\) and \(b\) are always opposite. When Alice measures \({x}_{1}\) and Bob measures\({y}_{0}\), there is no such correlation, and the probabilities of \(b\) to get + 1 and − 1 are equal. In general cases, we can define the probability distribution of measurement outcomes. Here, we use \(p\left(ab|xy\right)\) to represent the joint conditional probability distribution of outcomes when the measurement axes are \(x\) and\(y\). (2) No matter how far Alice and Bob are from each other, their measurement results always show correlation, that is as follows: