Chapter 7 discusses the Bell theorem and its demonstration using Mermin’s machine. The Bell theorem proves that a theory based on local realism (one with hidden variables) must satisfy certain inequalities; quantum mechanics predicts these are violated. Mermin designed a gedanken experiment having a double-particle source and two detectors each having three settings, which can correspond to measurements along three spin orientations. A simple hidden-variable assumption is shown to violate quantum predictions. The CHSH inequality, a form of the general Bell inequality, is proved. This relation has been shown to be violated experimentally in quantum experiments, implying that quantum mechanics indeed has “spooky action at a distance.” Many experiments over the years had loopholes, ways used to explain away the data in order to preserve local reality, but recent experiments have been able to eliminate most of these.
Chapter 7 discusses the Bell theorem and its demonstration using Mermin’s machine. The Bell theorem proves that a theory based on local realism (one with hidden variables) must satisfy certain inequalities; quantum mechanics predicts these are violated. Mermin designed a gedanken experiment having a double-particle source and two detectors each having three settings, which can correspond to measurements along three spin orientations. A simple hidden-variable assumption is shown to violate quantum predictions. The CHSH inequality, a form of the general Bell inequality, is proved. This relation has been shown to be violated experimentally in quantum experiments, implying that quantum mechanics indeed has “spooky action at a distance.” Many experiments over the years had loopholes, ways used to explain away the data in order to preserve local reality, but recent experiments have been able to eliminate most of these.
Bell's theorem is an important milestone in the development of quantum mechanics. It draws a line between concepts such as realism and locality on the one hand and quantum mechanics on the other. Experimentally veried violations of Bell inequalities can yield important information about what assumptions can be made about physical reality. Experiments have been conducted for over 40 years, but have left loopholes open in their executions. It is of great interest to close these loopholes, and in order to do so, further experiments are needed. The objective of this project is to investigate the underlying, previous work in the eld and to formulate a proposal for a new experiment. To do this, the derivations of the original Bell inequality and the CHSH inequality are presented along with previously conducted experiments. Also presented is the EPR paradox from which sprung the original Bell inequality.
Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.
Abstract The experimentally verified violation of Bell’s inequalities apparently implies that at least one of two beliefs, which are accepted as true in almost all scientific practice, must be false. These beliefs are: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. For many years, many theories have been proposed to reconcile the violation of Bell’s inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that Bell’s inequalities are equivalent to the conditions to decide the completeness of a theory according to Boolean logic. Therefore, any theory aimed to violate Bell’s inequalities must start by giving up Boolean logic. Two problems are therefore defined: the “soft” one is to explain the violation of Bell’s inequalities without violating (non-Boolean) Local Realism; the “hard” one is to predict the time values when single particles are detected, in such a way that the resulting number of coincidences violates Bell’s inequalities. A simple hidden variables model is introduced, which solves the “soft” problem even in an ideal setup. This is possible thanks to the use of vectors in real space as the hidden variables and the corresponding operation (projection), which do not hold to Boolean logic. This model reconciles the violation of Bell’s inequalities with Local Realism and should end a controversy that has lasted for decades. Regarding the “hard” problem, the introduced model is as incomplete as Quantum Mechanics is. It is claimed that solving the “hard” problem requires devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variable values as input data and to replace the statistical Born’s rule of Quantum Mechanics with a deterministic threshold condition.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
The recent loophole-free experiments have confirmed the violation of Bell’s inequalities in nature. Yet, in order to insert measured values in Bell’s inequalities, it is unavoidable to make a hypothesis similar to “ergodicity at the hidden variables level”. This possibility opens a promising way out from the old controversy between quantum mechanics and local realism. Here, I review the reason why such a hypothesis (actually, it is one of a set of related hypotheses) in addition to local realism is necessary, and present a simple example, related to Bell’s inequalities, where the hypothesis is violated. This example shows that the violation of the additional hypothesis is necessary, but not sufficient, to violate Bell’s inequalities without violating local realism. The example also provides some clues that may reveal the violation of the additional hypothesis in an experiment.
Quantum computers have tremendous potential of simulating large quantum systems that classical computers can’t. In the theoretical verification of quantum non-locality principle, Bell’s and Mermin’s inequalities stood out, which proved to be valid in multiple empirical experiments. In this paper we report the tests of Bell and Mermin’s inequalities carried out on the 2- qubit and 3-qubit IBM quantum computer respectively. By comparing the result we obtained and the pattern predicted by the inequalities, our experimental results indicate clear violations of the two inequalities; thereby, we validate postulates in quantum mechanics and disprove the local realism argument in classical hidden variable theories proposed by Einstein, Podolsky, and Rosen on quantum circuits.