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2022 Nobel Prize in physics: Bell inequalities and quantum entanglement

writerShang-Shu Li and Heng Fan

Vol.33 (Feb) 2023 | Article no.6-1 2023

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:

\(p\left(ab|xy\right)\ne p\left(a|x\right)p\left(b|y\right)\)

In the previous section, we mentioned that if the measurement depends on some hidden variable, the results will show a correlation. Let us think about the fact that we have a variable \(\lambda\) in addition. We cannot observe it for some physical reason, and the value of \(\lambda\) can vary in each measurement. Suppose it also satisfies a probability distribution \(q\left(lambda \right)\). The measurement distribution of Alice and Bob is now determined by two variables, which are denoted by \(p\left(a|x,lambda \right)\) and \(p\left(b|y,lambda \right),\) respectively. What does the joint probability look like? If we introduce the local condition, the results of Alice and Bob cannot affect each other, and then, the joint probability must be the product form as follows:

\(p\left(ab|xy,\lambda \right)=p\left(a|x,lambda \right)p\left(b|y,lambda \right)\)

Note that under this assumption, due to our ignorance of the hidden variable \(\lambda\), the observation results average over possible values of \(\lambda\); the observation results can then be correlated, i.e., as follows:

\(\begin{aligned}p\left(ab|xy\right)&={\int}_{\lambda }q\left(\lambda \right)p\left(a|x,\lambda \right)p\left(b|y,\lambda \right)\\ &\ne {\int }_{\lambda }q\left(\lambda \right)p\left(a|x,\lambda \right){\int }_{\lambda }q\left(\lambda \right)p\left(b|y,\lambda \right)\\ &=p\left(a|x\right)p\left(b|y\right)\end{aligned}\)

Equation (\(2\)) is the joint probability distribution obtained under the local hidden variable theory. It can be seen that Alice’s (Bob’s) measurement results only depend on the local variable \(x\)(\(y)\) and the hidden variable \(\lambda\). However, in quantum mechanics, Alice’s measurement results are correlated with that of Bob in some way, so that the joint probability of quantum mechanics may be beyond the expression ability of formula (\(2\)). Now let us try to figure out the limitation of the expression under localize hidden variable theory, which is Bell’s inequality.

Let us consider the relationship between the expectations of measurement results, under specified measurement axes. Define the correlation function as follows:

\(E\left({a}_{x},{b}_{y}\right)= \sum_{a,b}p\left(ab|xy\right)ab\)

Furthermore, define the observable \(S=E\left({a}_{0},{b}_{0}\right)+E\left({a}_{0}{b}_{1}\right)+E\left({a}_{1},{b}_{0}\right)-E\left({a}_{1},{b}_{1}\right)\). Due to the existence of formula (\(2\)), formula (\(3\)) can be written as the sum of product of local expectation value, under the distribution of implicit variable, namely \(E\left({a}_{x},{b}_{y}\right)={\int }_{lambda }\mathrm{q}\left(lambda \right)\mathrm{E}\left({a}_{\mathrm{x}};lambda \right)E\left({b}_{y};lambda \right)\). The local expectation \(\mathrm{E}\left({a}_{x};lambda \right)={\sum }_{a}a p\left(a|x,lambda \right)\) is in the interval [− 1, 1], the same for \(E\left({b}_{y};lambda \right)\).

Now, we have the following:

\(\begin{aligned}S&=\int \mathrm{d}\lambda \mathrm{q}\left(lambda \right)\left[\mathrm{E}\left({a}_{0};lambda \right)\left(E\left({b}_{0};lambda \right)+E\left({b}_{1};lambda \right)\right)+\mathrm{E}\left({a}_{1};lambda \right)\left(E\left({b}_{0};lambda \right)-E\left({b}_{1};lambda \right)\right)\right]\\ &\le \int \mathrm{d}\lambda \mathrm{q}\left(lambda \right)\left|E\left({b}_{0};lambda \right)+E\left({b}_{1};lambda \right)\right|+\left|E\left({b}_{0};lambda \right)-E\left({b}_{1};lambda \right)\right|\\ &\le \int \mathrm{d}\lambda \mathrm{q}\left(lambda \right)2\le 2\end{aligned}\)

We then obtain the famous CHSH inequality,

\(S=E\left({a}_{0},{b}_{0}\right)+E\left({a}_{0}{b}_{1}\right)+E\left({a}_{1},{b}_{0}\right)-E\left({a}_{1},{b}_{1}\right)\le 2\)

It is a variation of the original Bell inequality. In other words, under the probability distribution obtained by the local hidden variable hypothesis, the value of the correlation function \(S\) must satisfy this inequality. If the actual measurement results violate this inequality, it means that the real world cannot be described by local hidden variable theory. So, does quantum mechanics violate this inequality? The answer is yes. For example, we prepare Alice and Bob’s particle pairs into the spin singlets \(\left|{psi }_{AB}\right.\rangle =1/\sqrt{2}(|01\rangle -|10\rangle\), where \(\left|0\right.\rangle\) and \(\left|1\right.\rangle\) are the eigenvectors of the Pauli operator \({sigma }_{z}\). For any particle, we can measure the eigenvalue in its quantized direction \(\overrightarrow{x} \cdot \overrightarrow{sigma }\) in the experiment. Let the measured direction of Alice be \(\overrightarrow{x}\) and Bob’s be \(\overrightarrow{y}\). Then, the value of the correlation function obtained by quantum mechanics is \(-\overrightarrow{x}\cdot \overrightarrow{y}\). Now, we use these results to test the CHSH inequality and let Alice’s two measurement directions be two orthogonal basis \({x}_{0}={\widehat{\mathrm{e}}}_{1},{x}_{1}={\widehat{\mathrm{e}}}_{2}\). Bob’s two measurement directions are also orthogonal but have an angle with Alice’s. If Bob selects the two directions as \({y}_{0}=-\left({\widehat{\mathrm{e}}}_{1}+{\widehat{\mathrm{e}}}_{2}\right)/\sqrt{2},{y}_{1}=\left(-{\widehat{\mathrm{e}}}_{1}+ {\widehat{\mathrm{e}}}_{2}\right)/\sqrt{2}\), the value of the correlation function in CHSH inequality can be given by simple vector multiplication as follows:


It follows that \(S=2\sqrt{2}>2\), meaning that quantum mechanics indeed violate the CHSH inequality.

1.4 Experimental verification

Although Bell inequality is such a simple mathematical formula, experimental verification remains challenging. First of all, it requires that the entangled states are prepared with high fidelity experimentally and separated at a sufficiently long distance. The experimenter needs to make sure that the transmission of information below or equal the speed of light is excluded when measuring the two particles. The second challenge is the ability to measure particles in different arbitrary directions, since only in certain directions quantum mechanics can violate Bell’s inequality. In addition, the particle detection efficiency of the detector will also affect the verification of Bell inequality. Therefore, historically, the verification of Bell’s inequality has been carried out in the process of constantly closing the loopholes.

In 1972, John F. Clauser and Stuart Freedman performed the first Bell experiment [5]. They used the cascade transition of calcium atoms to produce entangled photon pairs. However, because the photon pair generation efficiency is very low, the measurement time reaches 200 h, and the distance between the two photons is too short; there is a loophole of locality. In addition, fixed measurement base is also one of the reasons for criticism.

In 1981 and 1982, Alain Aspect and his collaborators conducted a series of experiments that improved the measurement accuracy and reduced the loopholes in the verification of Bell’s inequality. In the first experiment [6], they used a double laser system to excite calcium atoms, producing pairs of entangled photons and improving the entanglement source. In the second experiment [7], a two-channel method was used to improve photon utilization. The measurement accuracy has been greatly improved. The third experiment [8] is the most important for closing the locality loophole. In the experiment, the two entangled photons are separated by about 12 m, and this distance needs 40 ns for the signal to travel at the speed of light. The distance between the photons and polarizer is 6 m. When performing measurement, polarizer rotates for no more than 20 ns. Using acousto-optical devices, photons can be switched to two measurement bases on even shorter time scales. The measurement time is much less than the time it takes for the signal to travel between two photons at the speed of light, closing the locality loophole.

In 1998, Anton Zeilinger’s team tested the Bell inequality under strict local conditions [9], with observers up to 400 m apart, closing the locality loophole completely. Subsequently, there have been a lot of experiments on the violation of Bell’s inequality. They are all aimed at closing the loopholes in the verification of quantum mechanics from various aspects, so that we are more and more confident in using quantum mechanics to describe the world. One interesting experiment is the Big Bell Test [10], which was designed to eliminate the effect of pseudo-randomness on the verification of Bell’s inequality. We know that random numbers generated by computers in simulations or experiments are pseudo-random numbers. As long as we give a certain seed, then the following series of random numbers are determined. This leads to the possibility that the correlation of experimental results measured in this way may exceed the limit represented by Bell’s inequality.

So how do you get a “true” random number? Big Bell Test proposes to solve this issue by using human’s free will to generate random numbers, as long as you believe that a person’s will is free and random. Okay, no, we do not believe it neither, maybe the experimenters have OCD (obsessive–compulsive disorder) or something related. To solve this problem, the researchers gathered more than 100,000 volunteers around the world and asked them to quickly and randomly press either 0 or 1 button in a game of tricks. They then uploaded the choices to the cloud and randomly sent them to different experimenters to use as random number generators for their experiments. Through the free will of a large number of participants, the Big Bell experiment closed the free choice loophole in a wider scope, strongly negating the localized hidden variable theory. So far, quantum mechanics has been almost perfectly demonstrated to be complete.

1.5 The second quantum revolution

Quantum mechanics has been verified to be correct, but some problems posed by non-locality need to be explained here. For example, can quantum entanglement transmit information faster than light? The answer is no; even though it looks like that the action at a distance travels faster than light, it does not transmit any information. Take the EPR pair as an example. First, since the collapse of the measurement is random, neither Alice nor Bob can encode the information into the EPR and decode it by measurement (without the help of classical means). Second, since there is no classical communication between Alice and Bob, the probability distribution of Alice’s measured results will not change regardless of whether Bob measures or not, namely regardless of whether Alice’s particles collapse. Therefore, Bob’s measurement operation will not transmit any information to Alice. In quantum mechanics, this is represented by the fact that the density matrix describing Alice’s particles does not change. So, does quantum entanglement make any sense? Of course, the violation of Bell’s inequality shows that entanglement is a kind of resource that transcends classical one, which indicates that even with infinite classical resources, we cannot achieve the results of quantum entanglement.

The research of such problems and all other efforts gave birth to the second quantum revolution which aims at the processing and application of quantum information. The first quantum mechanical revolution in history happens after the establishment of quantum mechanics. In this revolution, various classical applications based on quantum principles were developed, such as laser, semiconductor, and nuclear energy, which enabled mankind to quickly step into the information age, while the second quantum mechanical revolution is aimed to directly develop the applications of quantum coherence and quantum entanglement in quantum mechanics. Quantum information technology takes quantum bits as the basic unit, and the generation, transmission, processing, and detection of quantum information all follow the principle of quantum mechanics. In the last decades, the development of quantum computing theory and applications of quantum communication have made us see the potential of quantum technology to change the world. On the one hand, the development of quantum technology is to use the principles of quantum mechanics to process, transfer, and calculate quantum information; on the other hand, it also deepens our understanding of quantum mechanics.

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[Source: https://link.springer.com/article/10.1007/s43673-023-00075-6]