Bell test

A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables (called "hidden" because they are not a feature of quantum theory) to explain the behavior of particles like photons and electrons. The test empirically evaluates the implications of Bell's theorem. , all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave. Many types of Bell tests have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests". Bell inequality violations are also used in some quantum cryptography protocols, whereby a spy's presence is detected when Bell's inequalities cease to be violated.

A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables (called "hidden" because they are not a feature of quantum theory) to explain the behavior of particles like photons and electrons. The test empirically evaluates the implications of Bell's theorem. , all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave. Many types of Bell tests have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests". Bell inequality violations are also used in some quantum cryptography protocols, whereby a spy's presence is detected when Bell's inequalities cease to be violated.

Overview

The Bell test has its origins in the debate between Einstein and other pioneers of quantum physics, principally Niels Bohr. One feature of the theory of quantum mechanics under debate was the meaning of Heisenberg's uncertainty principle. This principle states that if some information is known about a given particle, there is some other information about it that is impossible to know. An example of this is found in observations of the position and the momentum of a given particle. According to the uncertainty principle, a particle's momentum and its position cannot simultaneously be determined with arbitrarily high precision. In 1935, Einstein, Boris Podolsky, and Nathan Rosen published a claim that quantum mechanics predicts that more information about a pair of entangled particles could be observed than Heisenberg's principle allowed, which would only be possible if information were travelling instantly between the two particles. This produces a paradox which came to be known as the "EPR paradox" after the three authors. It arises if any effect felt in one location is not the result of a cause that occurred in its past light cone, relative to its location. This action at a distance seems to violate causality, by allowing information between the two locations to travel faster than the speed of light. However, it is a common misconception to think that any information can be shared between two observers faster than the speed of light using entangled particles; the hypothetical information transfer here is between the particles. See no-communication theorem for further explanation. Based on this, the authors concluded that the quantum wave function does not provide a complete description of reality. They suggested that there must be some local hidden variables at work in order to account for the behavior of entangled particles. In a theory of hidden variables, as Einstein envisaged it, the randomness and indeterminacy seen in the behavior of quantum particles would only be apparent. For example, if one knew the details of all the hidden variables associated with a particle, then one could predict both its position and momentum. The uncertainty that had been quantified by Heisenberg's principle would simply be an artifact of not having complete information about the hidden variables. Furthermore, Einstein argued that the hidden variables should obey the condition of locality: Whatever the hidden variables actually are, the behavior of the hidden variables for one particle should not be able to instantly affect the behavior of those for another particle far away. This idea, called the principle of locality, is rooted in intuition from classical physics that physical interactions do not propagate instantly across space. These ideas were the subject of ongoing debate between their proponents. In particular, Einstein himself did not approve of the way Podolsky had stated the problem in the famous EPR paper. In 1964, John Stewart Bell proposed his famous theorem, which states that no physical theory of hidden local variables can ever reproduce all the predictions of quantum mechanics. Implicit in the theorem is the proposition that the determinism of classical physics is fundamentally incapable of describing quantum mechanics. Bell expanded on the theorem to provide what would become the conceptual foundation of the Bell test experiments. A typical experiment involves the observation of particles, often photons, in an apparatus designed to produce entangled pairs and allow for the measurement of some characteristic of each, such as their spin. The results of the experiment could then be compared to what was predicted by local realism and those predicted by quantum mechanics. In theory, the results could be "coincidentally" consistent with both. To address this problem, Bell proposed a mathematical description of local realism that placed a statistical limit on the likelihood of that eventuality. If the results of an experiment violate Bell's inequality, local hidden variables can be ruled out as their cause. Later researchers built on Bell's work by proposing new inequalities that serve the same purpose and refine the basic idea in one way or another. Consequently, the term "Bell inequality" can mean any one of a number of inequalities satisfied by local hidden-variables theories; in practice, many present-day experiments employ the CHSH inequality. All these inequalities, like the original devised by Bell, express the idea that assuming local realism places restrictions on the statistical results of experiments on sets of particles that have taken part in an interaction and then separated. To date, all Bell tests have supported the theory of quantum physics, and not the hypothesis of local hidden variables. These efforts to experimentally validate violations of the Bell inequalities resulted in John Clauser, Alain Aspect, and Anton Zeilinger being awarded the 2022 Nobel Prize in Physics. {{NumBlk|:| E = \frac {N_{++} - N_{+-} - N_{-+} + N_{--}} {N_{++} + N_{+-} + N_{-+}+ N_{--}} | }} Once all four E 's have been estimated, an experimental estimate of the test statistic can be found. If S is numerically greater than 2 it has infringed the CHSH inequality. The experiment is declared to have supported the QM prediction and ruled out all local hidden-variable theories. A strong assumption has had to be made, however, to justify use of expression (2), namely, that the sample of detected pairs is representative of the pairs emitted by the source. Denial of this assumption is called the fair sampling loophole. The first published experiment by Hensen et al. used a photonic link to entangle the electron spins of two nitrogen-vacancy defect centres in diamonds 1.3 kilometers apart and measured a violation of the CHSH inequality (S = 2.42 ± 0.20). Thereby the local-realist hypothesis could be rejected with a p-value of 0.039. Both simultaneously published experiments by Giustina et al. and Shalm et al. used entangled photons to obtain a Bell inequality violation with high statistical significance (p-value ≪10 −6 ). Notably, the experiment by Shalm et al. also combined three types of (quasi-)random number generators to determine the measurement basis choices. One of these methods, detailed in an ancillary file, is the "'Cultural' pseudorandom source" which involved using bit strings from popular media such as the Back to the Future films, Star Trek: Beyond the Final Frontier, Monty Python and the Holy Grail, and the television shows Saved by the Bell and Dr. Who. Physicists at LMU of Munich and the Max Planck Institute of Quantum Optics published results from an experiment in which they observed a Bell inequality violation using entangled spin states of two atoms with a separation distance of 398 meters in which the detection loophole, the locality loophole, and the memory loophole were closed. The violation of S = 2.221 ± 0.033 rejected local realism with a significance value of P = 1.02×10 −16 when taking into account 7 months of data and 55000 events or an upper bound of P = 2.57×10 −9 from a single run with 10000 events.

Detection loophole

A common problem in optical Bell tests is that only a small fraction of the emitted photons are detected. It is then possible that the correlations of the detected photons are unrepresentative: although they show a violation of a Bell inequality, if all photons were detected the Bell inequality would actually be respected. This was first noted by Philip M. Pearle in 1970, who devised a local hidden variable model that faked a Bell violation by letting the photon be detected only if the measurement setting was favourable. The assumption that this does not happen, i.e., that the small sample is actually representative of the whole is called the fair sampling assumption. To do away with this assumption it is necessary to detect a sufficiently large fraction of the photons. This is usually characterized in terms of the detection efficiency \eta , defined as the probability that a photodetector detects a photon that arrives at it. Anupam Garg and N. David Mermin showed that when using a maximally entangled state and the CHSH inequality an efficiency of \eta > 2\sqrt2-2 \approx 0.83 is required for a loophole-free violation. Later Philippe H. Eberhard showed that when using a partially entangled state a loophole-free violation is possible for \eta > 2/3 \approx 0.67 , which is the optimal bound for the CHSH inequality. Other Bell inequalities allow for even lower bounds. For example, there exists a four-setting inequality which is violated for \eta > (\sqrt5-1)/2 \approx 0.62 . Historically, only experiments with non-optical systems have been able to reach high enough efficiencies to close this loophole, such as trapped ions, superconducting qubits, and nitrogen-vacancy centers. These experiments were not able to close the locality loophole, which is easy to do with photons. More recently, however, optical setups have managed to reach sufficiently high detection efficiencies by using superconducting photodetectors, and hybrid setups have managed to combine the high detection efficiency typical of matter systems with the ease of distributing entanglement at a distance typical of photonic systems.

Locality loophole

One of the assumptions of Bell's theorem is the one of locality, namely that the choice of setting at a measurement site does not influence the result of the other. The motivation for this assumption is the theory of relativity, that prohibits communication faster than light. For this motivation to apply to an experiment, it needs to have space-like separation between its measurement events. That is, the time that passes between the choice of measurement setting and the production of an outcome must be shorter than the time it takes for a light signal to travel between the measurement sites. The first experiment that strived to respect this condition was Aspect's 1982 experiment. In it the settings were changed fast enough, but deterministically. The first experiment to change the settings randomly, with the choices made by a quantum random number generator, was Weihs et al.'s 1998 experiment. Scheidl et al. improved on this further in 2010 by conducting an experiment between locations separated by a distance of .

Coincidence loophole

In many experiments, especially those based on photon polarization, pairs of events in the two wings of the experiment are only identified as belonging to a single pair after the experiment is performed, by judging whether or not their detection times are close enough to one another. This generates a new possibility for a local hidden variables theory to "fake" quantum correlations: delay the detection time of each of the two particles by a larger or smaller amount depending on some relationship between hidden variables carried by the particles and the detector settings encountered at the measurement station. The coincidence loophole can be ruled out entirely simply by working with a pre-fixed lattice of detection windows which are short enough that most pairs of events occurring in the same window do originate with the same emission and long enough that a true pair is not separated by a window boundary.

Memory loophole

In most experiments, measurements are repeatedly made at the same two locations. A local hidden variable theory could exploit the memory of past measurement settings and outcomes in order to increase the violation of a Bell inequality. Moreover, physical parameters might be varying in time. It has been shown that, provided each new pair of measurements is done with a new random pair of measurement settings, that neither memory nor time inhomogeneity have a serious effect on the experiment.

Superdeterminism

A necessary assumption to derive Bell's theorem is that the hidden variables are not correlated with the measurement settings. This assumption has been justified on the grounds that the experimenter has "free will" to choose the settings, and that such is necessary to do science in the first place. A (hypothetical) theory where the choice of measurement is determined by the system being measured is known as superdeterministic.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1365877846 under CC BY-SA 4.0; modifications have been made.