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Entropy

Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism

Alejandro Andrés Hnilo

Published Apr 22, 2017• 5 citations
Publisher

Abstract

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.

DOI
10.3390/e19040180

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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.
The Bell's inequalities are derived from the hypotheses of Locality, Realism and (what is lesser known) the equality between the factual and the counterfactual time averages of the expectation values of observables. The necessity of a hypothesis additional to Local Realism opens a promising way out to the old controversy between Quantum Mechanics and Local Realism. For, it is possible to speculate that it is this additional hypothesis, and not Local Realism, what is disproved in the experiments reporting the violation of the Bell's inequalities. Yet, there are doubts on how the additional hypothesis may be violated in a physically reasonable process. A simple example showing that this is possible, the relationship between the validity of the additional hypothesis and the validity of the Bell's inequalities, and considerations on its physical meaning, are presented.
For almost three decades in the twentieth century, the physics community believed that John von Neumann had proved the impossibility of completing quantum mechanics by a local realist, hidden-variables theory. Although Grete Hermann had raised strong objections to von Neumann's proof, she was largely ignored. This situation lasted, until John Bell rediscovered that von Neumann's proof was flawed: a \emph{sufficient} condition for local realism had been taken as a \emph{necessary} one. Bell subsequently established various constraints on hidden-variables theories, in the form of inequalities that can be submitted to experimental test. All performed tests to date have opened some loopholes. The quest to close them motivated great technical achievements and ongoing efforts to improve what has already been reached. There is, however, a rather ironic twist concerning Bell inequalities. On deriving them, Bell also took a sufficient condition for local-realism, as if it were a necessary one. As a consequence, even completely loophole-free Bell inequality violations would not disprove local realism. We argue that Bell inequalities cannot follow from local-realism alone. The proof is given by constructing three local-realist models that entail Bell inequality violations.
Assuming that there is no way of sending signals propagating faster than light and that free will exists, the loophole-free observed violation of Bell's inequalities demonstrates that at least one of three fundamental hypotheses involved in the derivation and observation of the inequalities is false: Locality, Realism, or Ergodicity. An experiment is proposed to obtain some evidence about which one is the false one. It is based on recording the time evolution of the rate of non-random series of outcomes that are generated in a specially designed Bell's setup. The results of such experiment would be important not only to the foundations of Quantum Mechanics, but they would also have immediate practical impact on the efficient use of quantum-based random number generators and the security of Quantum Key Distribution using entangled states.
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.
Abstract. Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or “loopholes ” in available experiment tests, local realist theories can reproduce the data from these experiments. This paper reviews such loopholes, what effect they have on Bell inequality tests, and how to avoid them in experiment. Avoiding all these simultaneously in one experiment, usually called a “loophole-free ” or “definitive ” Bell test, remains an open task, but is very important for technological tasks such as device-independent security of quantum cryptography, and ultimately for our understanding of the world. ar X iv
Bell's theorem shows that local realistic theories place strong restrictions on observable correlations between different systems, giving rise to Bell's inequality which can be violated in experiments using entangled quantum states. Bell's theorem is based on the assumptions of realism, locality, and the freedom to choose between measurement settings. In experimental tests, "loopholes" arise which allow observed violations to still be explained by local realistic theories. Violating Bell's inequality while simultaneously closing all such loopholes is one of the most significant still open challenges in fundamental physics today. In this paper, we present an experiment that violates Bell's inequality while simultaneously closing the locality loophole and addressing the freedom-of-choice loophole, also closing the latter within a reasonable set of assumptions. We also explain that the locality and freedom-of-choice loopholes can be closed only within nondeterminism, i.e., in the context of stochastic local realism.