Bell test

Scheme of a "two-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.

The diagram shows a typical optical experiment of the two-channel kind for which Alain Aspect set a precedent in 1982.[8] Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.

Four separate subexperiments are conducted, corresponding to the four terms E(a, b) in the test statistic S (equation (2) shown below). The settings a, a′, b and b′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality.

For each selected value of a and b, the numbers of coincidences in each category (N₊₊, N₋₋, N₊₋ and N₋₊) are recorded. The experimental estimate for E(a, b) is then calculated as:

${\displaystyle E={\frac {N_{++}-N_{+-}-N_{-+}+N_{--}}{N_{++}+N_{+-}+N_{-+}+N_{--}}}}$

(1)

Once all four E’s have been estimated, an experimental estimate of the test statistic

${\displaystyle S=E(a,b)-E\left(a,b'\right)+E\left(a',b\right)+E\left(a',b'\right).}$

(2)

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). It has been assumed that the sample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be true comprises the fair sampling loophole.

The derivation of the inequality is given in the CHSH Bell test page.

Setup for a "single-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a single channel (e.g. "pile of plates") polariser whose orientation can be set by the experimenter. Emerging signals are detected and coincidences counted by the coincidence monitor CM.

Prior to 1982 all actual Bell tests used "single-channel" polarisers and variations on an inequality designed for this setup. The latter is described in Clauser, Horne, Shimony and Holt's much-cited 1969 article as being the one suitable for practical use.[5] As with the CHSH test, there are four subexperiments in which each polariser takes one of two possible settings, but in addition there are other subexperiments in which one or other polariser or both are absent. Counts are taken as before and used to estimate the test statistic.

${\displaystyle S={\frac {N(a,b)-N(a,b')+N(a',b)+N(a',b')-N(a',\infty )-N(\infty ,b)}{N(\infty ,\infty )}},}$

(3)

where the symbol ∞ indicates absence of a polariser.

If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted local realism". In order to derive (3), CHSH in their 1969 paper had to make an extra assumption, the so-called "fair sampling" assumption. This means that the probability of detection of a given photon, once it has passed the polarizer, is independent of the polarizer setting (including the 'absence' setting). If this assumption were violated, then in principle a local hidden-variable (LHV) model could violate the CHSH inequality.

In a later 1974 article, Clauser and Horne replaced this assumption by a much weaker, "no enhancement" assumption, deriving a modified inequality, see the page on Clauser and Horne's 1974 Bell test.[9]

Stuart J. Freedman and John Clauser carried out the first actual Bell test, using Freedman's inequality, a variant on the CH74 inequality.[11]

Alain Aspect and his team at Orsay, Paris, conducted three Bell tests using calcium cascade sources. The first and last used the CH74 inequality. The second was the first application of the CHSH inequality. The third (and most famous) was arranged such that the choice between the two settings on each side was made during the flight of the photons (as originally suggested by John Bell).[12][13]

The Geneva 1998 Bell test experiments showed that distance did not destroy the "entanglement". Light was sent in fibre optic cables over distances of several kilometers before it was analysed. As with almost all Bell tests since about 1985, a "parametric down-conversion" (PDC) source was used.[14][15]

In 1998 Gregor Weihs and a team at Innsbruck, led by Anton Zeilinger, conducted an ingenious experiment that closed the "locality" loophole, improving on Aspect's of 1982. The choice of detector was made using a quantum process to ensure that it was random. This test violated the CHSH inequality by over 30 standard deviations, the coincidence curves agreeing with those predicted by quantum theory.[16]

This is the first of new Bell-type experiments on more than two particles; this one uses the so-called GHZ state of three particles.[17]

The detection loophole was first closed in an experiment with two entangled trapped ions, carried out in the ion storage group of David Wineland at the National Institute of Standards and Technology in Boulder. The experiment had detection efficiencies well over 90%.[18]

Using semileptonic B0 decays of Υ(4S) at Belle experiment, a clear violation of Bell Inequality in particle-antiparticle correlation is observed.[19]

A specific class of non-local theories suggested by Anthony Leggett is ruled out. Based on this, the authors conclude that any possible non-local hidden-variable theory consistent with quantum mechanics must be highly counterintuitive.[20][21]

This experiment filled a loophole by providing an 18 km separation between detectors, which is sufficient to allow the completion of the quantum state measurements before any information could have traveled between the two detectors.[22][23]

This was the first experiment testing Bell inequalities with solid-state qubits (superconducting Josephson phase qubits were used). This experiment surmounted the detection loophole using a pair of superconducting qubits in an entangled state. However, the experiment still suffered from the locality loophole because the qubits were only separated by a few millimeters.[24]

The detection loophole for photons has been closed for the first time by Marissa Giustina, using highly efficient detectors. This makes photons the first system for which all of the main loopholes have been closed, albeit in different experiments.[25][26]

The Christensen et al. (2013)[27] experiment is similar to that of Giustina et al.[25] Giustina et al. did just four long runs with constant measurement settings (one for each of the four pairs of settings). The experiment was not pulsed so that formation of "pairs" from the two records of measurement results (Alice and Bob) had to be done after the experiment which in fact exposes the experiment to the coincidence loophole. This led to a reanalysis of the experimental data in a way which removed the coincidence loophole, and fortunately the new analysis still showed a violation of the appropriate CHSH or CH inequality.[26] On the other hand, the Christensen et al. experiment was pulsed and measurement settings were frequently reset in a random way, though only once every 1000 particle pairs, not every time.[27]

In 2015 the first three significant-loophole-free Bell-tests were published within three months by independent groups in Delft, Vienna and Boulder. All three tests simultaneously addressed the detection loophole, the locality loophole, and the memory loophole. This makes them “loophole-free” in the sense that all remaining conceivable loopholes like superdeterminism require truly exotic hypotheses that might never get closed experimentally.

The first published experiment by Hensen et al.[10] 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.[28] and Shalm et al.[29] 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.[30]

Using a witness for Bell correlations derived from a multi-partite Bell inequality, physicists at the University of Basel were able to conclude for the first time Bell correlation in a many-body system composed by about 480 atoms in a Bose-Einstein condensate. Even though loopholes were not closed, this experiment shows the possibility of observing Bell correlations in the macroscopic regime. [31]

Physicists led by David Kaiser of the Massachusetts Institute of Technology and Anton Zeilinger of the Institute for Quantum Optics and Quantum Information and University of Vienna performed an experiment that "produced results consistent with nonlocality" by measuring starlight that had taken 600 years to travel to Earth.[32] The experiment “represents the first experiment to dramatically limit the space-time region in which hidden variables could be relevant.”[33][34][35]

Physicists at the Ludwig Maximilian University 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.[36]

An international collaborative scientific effort showed that human free will can be used to close the 'freedom-of-choice loophole'. This was achieved by collecting random decisions from humans instead of random number generators. Around 100,000 participants were recruited in order to provide sufficient input for the experiment to be statistically significant.[37]

In 2018, an international team used light from two quasars (one whose light was generated approximately eight billion years ago and the other approximately twelve billion years ago) as the basis for their measurement settings.[38] This experiment pushed the timeframe for when the settings could have been mutually determined to at least 7.8 billion years in the past, a substantial fraction of the superdeterministic limit (that being the creation of the universe 13.8 billion years ago).[39]

The 2019 PBS Nova episode Einstein's Quantum Riddle documents this "cosmic Bell test" measurement, with footage of the scientific team on-site at the high-altitude Teide Observatory located in the Canary Islands.[40]

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 Pearle in 1970,[46] 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 ${\displaystyle \eta }$, defined as the probability that a photodetector detects a photon that arrives at it. Garg and Mermin showed that when using a maximally entangled state and the CHSH inequality an efficiency of ${\displaystyle \eta >2{\sqrt {2}}-2\approx 0.83}$ is required for a loophole-free violation.[47] Later Eberhard showed that when using a partially entangled state a loophole-free violation is possible for ${\displaystyle \eta >2/3\approx 0.67}$,[48] which is the optimal bound for the CHSH inequality.[49] Other Bell inequalities allow for even lower bounds. For example, there exists a four-setting inequality which is violated for ${\displaystyle \eta >({\sqrt {5}}-1)/2\approx 0.62}$.[50]

Historically, only experiments with non-optical systems have been able to reach high enough efficiencies to close this loophole, such as trapped ions,[51] superconducting qubits,[52] and nitrogen-vacancy centers.[53] 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,[28][29] 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.[10]

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 measurements 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.[54]

The first experiment that strived to respect this condition was Alain Aspect's 1982 experiment.[13] 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.[16] Scheidl et al. improved on this further in 2010 by conducting an experiment between locations separated by a distance of 144 km (89 mi).[55]

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.[56]

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.[56]

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.[57][58][59]

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 it 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.[60]