1. Local realism and Bell's inequality
Einstein and two younger colleagues, Boris Podolsky and Nathan Rosen, found that quantum entanglement conflicts with local realism, arguing that quantum mechanics is incomplete [1,2]. This means that in addition to the quantum states in quantum mechanics, there are additional variables in the physical system that characterize the exact state of the system. These additional variables are called hidden variables, and they represent what is known as realism. If a theory that replaces quantum mechanics contains hidden variables, it is called hidden variable theory. If this theory also satisfies locality, it is called local hidden variable theory, or local realism.
Prior to the EPR paper, von Neumann had mathematically demonstrated that hidden variables did not exist in 1931 [3]. After the EPR paper, there was some discussion of hidden variable theory in the 1950s and 1960s, particularly a series of work by David Bohm. John Bell pointed out in 1964 (published in 1966) that von Neumann's proof did not hold [4].
In 1964, Bell proposed that local realism is contradictory to quantum mechanics, and he published an inequality that any theory of local hidden variables should satisfy.[5] Later all inequalities of this class were called Bell's inequalities , which were related to the measurement results of two subsystems , each measured by a local observer. The correlation of various measurement results is calculated using the theory of local hidden variables, the result of which satisfies Bell's inequality, and in quantum mechanics, if these two subsystems are described in terms of some quantum entanglement state, then the result of the calculation according to quantum mechanics violates Bell's inequality.
In the paper, entitled "On the Einstein-Podolsky-Rosen Paradox," Bell uses a form pioneered by Bohm, based on spin
language, but also suitable for other similar discrete variables, such as photon polarization. The measurements A and B of the two particles are the spin values divided by the appropriate coefficients, both of which are 1 or -1, depending on the hidden variable and the respective measurement directions a and b, so A(a,λ) = ±1, B(b,λ) = ±1. According to local realism , their association is P(a,b) = ∫dλρ(λ)A(a,λ)B(b,λ). For A and B strictly correlating A(a,λ) = -B(a,λ), Bell proved
1+P(b,c)≥| P(a,b)–P(a,c)|.
And for spin-entangled states
, quantum mechanics gives P(a,b) = –a•b, selects the appropriate a, b, c, and obtains the violation of the inequality. Quantum entangled states for photon polarization
, P(a,b) = -cos2θ, where θ is the angle between a and b.
The fundamental problems of quantum mechanics were once regarded as "just philosophy", and Bell's inequality shows that this is physics, which has a theoretical and experimental taste, transforming the original metaphysical discussion into a judgment that can be determined quantitatively by experiments, and transforming philosophical problems into quantitative scientific problems.
The experiment to test whether nature satisfies Bell's inequality is called the Bell test. Bell testing requires the use of subsystems that live apart and are in a quantum entangled state, as well as rapid and efficient detection, and the independent arrangement of each measuring device that is unpredictable in advance. All work on Bell's inequality violations (or Bell's theorem) builds on Bell's seminal work.
The experiment judged that quantum mechanics won and local realism failed. However, for a long time, there were logical loopholes or additional assumptions in experimental judgments, and it was only in recent years that they were basically eliminated. The formulation and verification of Bell's inequality is closely related to the rise of quantum informatics, including concepts and experimental techniques. The work of the 2022 Nobel Prize in Physics is a major contribution to both aspects.
2. Bell-CHSH inequality and experiment
2.1 CHSH inequality
The specific form of Bell's original inequality relied on assumptions that were too idealistic, such as strict associations, to be verified in actual experiments, and therefore unsuitable for real experiments. In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt generalized Bell's inequality, often referred to as CHSH or Bell-CHSH inequality.[6]
The Bell-CHSH inequality is more suitable for real-world situations and can be tested in real-world experiments.
Continuing our discussion of Bell's inequality above. Consider A(a,λ)[B(b,λ)+B(b',λ)]+A(a',λ)[B(b,λ)-B(b',λ)]. It must be equal to ±2, because B(b,λ)+B(b',λ) and B(b,λ)-B(b',λ) must have one equal to ±2 and one equal to 0. This gives S=P(a,b)+P(a,b')+P(a',b)-P(a',b)-P(a',b') satisfied
–2≤S≤2.
This is the Bell-CHSH inequality. For the Bell state, for example, S= can be reached
, which violates the Bell-CHSH inequality.
Therefore, as long as there is local realism, the Bell-CHSH inequality can be held and can be tested experimentally. And quantum mechanics violates it. Therefore, which is correct between quantum mechanics and local realism depends on which one is consistent with experimentation.
In addition, in 1989, Anton Zeilinger, together with Daniel Greenberg and Michael Horne, discovered that a three-particle quantum entangled state has special properties that do not require statistical averaging or construction inequalities, which conflicts with local realism [7].
2.2 Freedman-Clauser experiment
When the Bell-CHSH inequality was proposed in 1969, Krause was a doctoral student in molecular astrophysics. After receiving his Ph.D. in 1970, he moved to the University of California, Berkeley, where he became a postdoc at Charles Townes, where he was allowed to study Bell's inequality autonomously. In Berkeley, in 1967, Eurgene Commins' student Carl Kocher's doctoral thesis work was to study the temporal association of photon pairs from the same atomic source [8].
In this system, cascading transitions produce entangled photon pairs. An outer electron of a calcium atom is removed from the ground state
Be stimulated to
, and jumped to
, jumping from this energy level to
, emits a photon; Then jump back to the ground state
, emits another photon. In order to keep the conservation of the universe symmetry as the even symmetry , the conservation of angular momentum is 0 , so the two-photon polarization state must be
。 In this entangled state, the coincidence of photons detected on both sides is
, where φ is the angle between the polarizers that detect two photons. But the polarizer angles between the two detection photons that Cork chose to study were 0 and 90 degrees, and could not be used to test Bell's inequality.
Krause and Cummins' doctoral student Stuart Freedman (now deceased) modified the experimental setup to improve the efficiency of the polarizer. In this system , the CHSH inequality is given
thereinto
is the compliance rate without a polarizer.
Krause and Friedman experimentally obtained that the left side of the above equation is 0.05±0.008, which violates Bell's inequality with an experimental accuracy of 6 standard deviations [9].
Schematic diagram of the cartoon of Krause's and Friedman's experiment (Image from nobelprize.org)
This initial experimental attempt had loopholes and limitations—the efficiency of generating and detecting particles was low, and the measurements were preset, so logically it was possible that hidden variables made the detection of particles selective, or that the setting of the measuring device (especially the measurement direction of the polarizer) affected the polarization of photons when emitted, resulting in a violation of Bell's inequality and not meeting local requirements.
Locality is a key premise of Bell's inequality. The measurements of the two separate subsystems must be independent of each other, including the choice of which measurement to make, such as position or momentum, or transverse or longitudinal magnetic moment (magnetic moment proportional to spin), or the direction of light transmission of the polarizer. Therefore, it must be ensured that the measurement time difference between the two is small enough that it is impossible for a physical signal to travel from one to the other. Because all signal velocities do not exceed the speed of light, experimentally it is necessary to ensure that the time difference measured by the two sides is less than the distance divided by the speed of light, in the language of relativity, which is called a space-like interval. The fixed setup of the Friedman-Krause experiment does not meet the local requirements.
2.3 Asper experiment
In 1981-1982, Alain Aspect, together with collaborators Phillipe Grangier, Gerard Roger and Jean Dalibard, conducted 3 experiments to observe a violation of the Bell-CHSH inequality with high precision, largely satisfying the local requirements.
In the first experiment [10], electrons are excited directly by two-photon absorption by two sets of lasers before the cascade process occurs
, which is better than before
Much more effective.
In the second experiment [11], measurements with a dual-channel polarizer yielded good statistics and a large violation of Bell's inequality, with an accuracy of tens of standard deviations.
Bell pointed out that the experimental setup had to be changed during the flight of the particles.[5] If the polarization measurement direction on both sides is randomly changed, and the time taken is less than the time for the two particles to reach the polarizer from the source, it can be ensured that when the entangled particles are measured, the space-time interval is empty, and the measurements on both sides have no causal correlation.
This was partially achieved in Aspe's third experiment. This is the most important of the three experiments [12]. The distance from the calcium atom to the polarizer is 6 meters, and the photon flight is only 20 nanoseconds, which is too late to rotate the polarizer during the photon flight. But they used the ingenious method devised by Aspe in earlier years.[13] A pair of photons passes through an acousto-optic switch before reaching a pair of polarizers and is directed to one of the two pairs of polarizers. The acousto-optic switches are switched every 10ns. The CHSH inequality used is given
, quantum mechanics gives 0.112.
The experiment yielded 0.101±0.020, which is consistent with quantum mechanics and violates the inequality. The accuracy is 5 standard deviations.
Caricature of Aspe's third experiment (image from nobelprize.org)
These experiments, and many later Bell test experiments, ruled that quantum mechanics won and local realism failed. However, there are still technical logical loopholes in these efforts, such as in detector efficiency or locality.
In Asper 's third experiment, because the polarizer's measurement direction changed after the entangled photon pair left the source, it is logically very likely that the polarization at the time of photon generation was not affected by the polarizer's measurement direction.
However, the distance between the two polarizers is very short, and due to technical limitations, it is not possible to randomly change the measurement device when the photon is flying. So in Aspe's third experiment, the changes in the measuring device were not random, but periodic. Specifically, I think that logically it is impossible to rule out a more complex "conspiracy theory": since the change of the measuring device is periodic, the relationship between the polarizer direction at the time of measurement and the polarizer direction in the past is determined, so the latter may also affect the polarization when the photon is generated. So Asper's third experiment doesn't close the local loophole, but it still has a historic milestone.
2.4 Salinger Group closes local vulnerabilities
In 1997, Salinger's experiments finally closed the local loophole [14]. In their experiment, the experimental setup that analyzed entangled photon pairs was 400 meters apart and traveled at the speed of light in 1300 nanoseconds. Entangled photon pairs are transmitted to the polarizer through the optical fiber. The orientation of the polarization analysis device for each photon has enough time to make a rapid random change—controlled by a random number generator and timed with an atomic clock. After measuring a series of photon pairs, the experimenters collected data from both sides and analyzed the correlation.
Schematic of the caricature of Salinger's group's Bell test experiment (image from nobelprize.org).
After closing the local vulnerability, Salinger's group did a number of Bell tests, one of which used wavelength information from photons from different quasars to determine the direction of polarization measurement.[15] This is reflected in the comics. At the same time, the experiments of Pan Jianwei's group used wavelength information from photons from different stars to determine the direction of polarization measurement [16].
In this experiment, various techniques improved a lot. It is worth pointing out that the type-II parametric down conversion method is used here to generate entangled photon pairs. This is a nonlinear optical process implemented with β-BBO (barium metaborate) crystals. β-BBO crystal is a UV-frequency doubling crystal first discovered and developed by the Fujian Institute of Structure of Matter, Chinese Academy of Sciences. This method produces entangled photon pairs that can be transmitted through optical fibers, thus separating large distances. This method was first developed by Ou Zeyu and L. Mandel, as well as Shi Yanhua and C. O. Alley implemented it in the 1980s and used it to do the Bell test[17,18], and later Ou Zeyu, Pereira, Kimble, and Peng Kunqi used this method to achieve entanglement of continuous variables (so-called light amplitude) and the Bell test [19]. Later, this method was used to achieve the Bell test of entangled photons separated by several kilometers [20] and tens of kilometers [21]. The use of this method by the Salinger Group began with a collaborative work with Shi Yanhua to achieve a Bell inequality violation of more than 100 standard deviations in 4 minutes [22].
As we said earlier, Einstein and others revealed the conflict between quantum entanglement and fixed-domain realism (that is, the combination of locality and realism), and Einstein is the biggest contributor to the study of quantum entanglement. But as Aspe once said, under the measurements chosen under relativistic separations (i.e., space-like separations), the violation of Bell's inequality means that Einstein images such as "explaining the association in terms of the common properties determined by the common source of particles and carried by photon pairs" can no longer be maintained, and we must conclude that entangled particle pairs are indeed inseparable wholes, and each particle cannot be given a separate local property [23].
3. Follow-up
In experiments on Bell's inequality, there have long been "detection holes". Because the entangled particles detected are only part of the initial entanglement pairs, how much of the detected particles are related to the experimental setup. Under the premise of fair sampling, the experimental statistical analysis can be used to test Bell's inequality. However, the efficiency of the detector is limited, and if the detection efficiency is not high enough, it may not be able to do fair sampling, which is to detect vulnerabilities.
To close the detection loophole and ensure fair sampling, the condition must be met: when a photon is measured on one side, the probability that a photon is also detected on the other side is greater than 2/3 [24]. Ion experiments in 2001 and 2008 [25,26] closed the detection loophole. In 2013, Salinger's group [27] and Kwiat's group [28] also closed detection loopholes in photon experiments.
In 2015, several experiments filled both local and detection vulnerabilities, the Salinger group [29] and NIST's Shalm group [30] both used rapidly changing polarizers and high-efficiency photon detectors, and Delft's Hensen group used two pairs of electron-photon pairs [31] to measure two photons, so that the two electrons were entangled. Later, in 2017, Weinfurter closed both loopholes with entangled atoms 398 meters apart.[32]
Let's take a look at the "free choice vulnerability". Bell's inequality is about the association between various measurement results of two subsystems, involving several different settings of the measuring device, such as the direction of measurement. This is completely free in the derivation of Bell's inequality and has nothing to do with hidden variables.
In the Bell test, these different settings need to be freely selected randomly. For a long time, in experiments, even if local vulnerabilities and detection vulnerabilities were filled, the arrangement of experimental devices was randomly selected by the instrument. This is not ideal, because what if the choices made by these instruments are themselves determined by hidden variables? This is called the "free choice loophole." Bell had proposed that the unpredictability of the arrangement of experimental devices could be guaranteed by the free choice of people. But the technology of the time couldn't do it.
On November 30, 2016, an experiment called The Big Bell Test was one such experiment, closing this "free choice loophole." The selections made in the experiment were from about 100,000 volunteers from around the world. For 12 hours, the volunteers generated 1,000 bits of data per second through an online game called "The BIG Bell Quest," for a total of 97347490 bits of data. Volunteers participating in the game were asked to input a certain random bit 0 or 1 within a certain amount of time, which was used to instruct the choices made in the experiment. A machine learning algorithm reminds volunteers to avoid predictability based on the bits entered, but not to select the resulting data.
12 laboratories on five continents did 13 Bell experiments in 12 hours. These experiments used data provided by 100,000 volunteers to arrange the measuring apparatus, and different experiments used different data. The results of the Bell test on different systems show that local realism is violated in these systems. Several of them are photon polarization experiments completed by Pan Jianwei's group and Salinger's group.
On May 9, 2018, Nature magazine published the results of these 13 Bell experiments under the title "Employment Choice Challenges Local Realism" [33,34], showing that local realism is violated in photons, single atoms, atomic ensembles and superconducting devices. This work represents another step forward in testing the fundamental theory of quantum mechanics.
Finally, since local realism conflicts with quantum mechanics, where does the source of the contradiction come from, localism or realism? To investigate this problem, Anthony J. Leggett considers a "cryptographic non-local realism": as nonlocality, for a given polarization direction, the measured quantity depends on both the measured polarizer direction and the polarizer direction on the other side. But the physical state is a statistical average of various polarization directions, obeying local laws such as Marius's law. In this regard, Leggett derived Leggett's inequality, which was violated by quantum mechanics [35]. Recently we proposed a generalized Leggett inequality, especially applicable to entangled mesons in particle physics, which is violated by quantum mechanics and particle physics [36].
The 2022 Nobel Prize in Physics was awarded to Krause, Asper and Salinger for their experiments on entangled photons, laying the groundwork for the violation of Bell's inequality and pioneering quantum information science. Their groundbreaking experiments made quantum entanglement a "powerful tool" that laid the foundation for a new era of quantum technology.
#娱兔迎春 #