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 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.
Conduct of optical Bell test experiments
In practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons, rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, the polarisation direction, though other properties can be used. Such experiments fall into two classes, depending on whether the analysers used have one or two output channels.A typical CHSH (two-channel) experiment
The diagram shows a typical optical experiment of the two-channel kind for which Alain Aspect set a precedent in 1982. Coincidences are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.Four separate subexperiments are conducted, corresponding to the four terms E in the test statistic S. 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 are recorded. The experimental estimate for E is then calculated as:
Once all four E
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, 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.
A typical CH74 (single-channel) experiment
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. 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.where the symbol ∞ indicates absence of a polariser.
If S exceeds 0 then the experiment is declared to have infringed the CH inequality and hence to have refuted local hidden-variables. This inequality is known as CH inequality instead of CHSH as it was also derived in a 1974 article by Clauser and Horne more rigorously and under weaker assumptions.
Experimental assumptions
In addition to the theoretical assumptions, there are practical ones. There may, for example, be a number of "accidental coincidences" in addition to those of interest. It is assumed that no bias is introduced by subtracting their estimated number before calculating S, but that this is true is not considered by some to be obvious. There may be synchronisation problems — ambiguity in recognising pairs because in practice they will not be detected at exactly the same time.Nevertheless, despite all the deficiencies of the actual experiments, one striking fact emerges: the results are, to a very good approximation, what quantum mechanics predicts. If imperfect experiments give us such excellent overlap with quantum predictions, most working quantum physicists would agree with John Bell in expecting that, when a perfect Bell test is done, the Bell inequalities will still be violated. This attitude has led to the emergence of a new sub-field of physics known as quantum information theory. One of the main achievements of this new branch of physics is showing that violation of Bell's inequalities leads to the possibility of a secure information transfer, which utilizes the so-called quantum cryptography.