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arXiv:2403.03236

Some consequences of Sica's approach to Bell's inequalities

Alejandro Andrés Hnilo

Published Mar 5, 2024• 1 citation
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Abstract

Louis Sica derived Bell's inequalities from the hypothesis that the time series of outcomes observed in one station does not change if the setting in the other (distant) station is changed. This derivation is based on arithmetical properties only. It does not involve the controversial definitions of Locality and Realism, it does not require the definition of probabilities, and is valid for series of any length. In this paper, Sica's approach is extended to series with non ideal efficiency and to the actual time structure of experimental data. The first extension leads to an interesting relationship, involving the entanglement parameter SCHSH and efficiency, that places the so-called 'detection loophole' under new light. The second extension makes visible that measuring with different settings unavoidably means recording series at different times. It replaces 'Local Realism' (as the assumption necessary for the validity of Bell's inequalities), with the assumption that the recorded series can be arbitrarily reordered. Violation of this latter assumption is, in my opinion, more acceptable to intuition than violation of Local Realism. The second extension also shows that the observation of a violation of Bell's inequalities implies that Sica's hypothesis is not valid, i.e., that the series in one station is different if the setting in the other station is changed. This result gives precise meaning to 'quantum non-locality', and also explains why it cannot be used for sending messages. Finally, it is demonstrated that a series of outcomes, even if it violates Bell's inequalities, can be always embedded in a set of factual and counter-factual data in which Sica's hypothesis is valid. In consequence, factual universe may be quantum (non-classical) or not, but the union of factual and counter-factual universes is always classical.

DOI
10.48550/arxiv.2403.03236

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Louis Sica derived Bell's inequalities from the hypothesis that the time series of outcomes observed in one station does not change if the setting in the other (distant) station is changed. This derivation is based on arithmetical properties only. It does not involve the controversial definitions of Locality and Realism, it does not require the definition of probabilities, and is valid for series of any length. In this paper, Sica's approach is extended to series with non ideal efficiency and to the actual time structure of experimental data. The first extension leads to an interesting relationship, involving the entanglement parameter SCHSH and efficiency, that places the so-called 'detection loophole' under new light. The second extension makes visible that measuring with different settings unavoidably means recording series at different times. It replaces 'Local Realism' (as the assumption necessary for the validity of Bell's inequalities), with the assumption that the recorded series can be arbitrarily reordered. Violation of this latter assumption is, in my opinion, more acceptable to intuition than violation of Local Realism. The second extension also shows that the observation of a violation of Bell's inequalities implies that Sica's hypothesis is not valid, i.e., that the series in one station is different if the setting in the other station is changed. This result gives precise meaning to 'quantum non-locality', and also explains why it cannot be used for sending messages. Finally, it is demonstrated that a series of outcomes, even if it violates Bell's inequalities, can be always embedded in a set of factual and counter-factual data in which Sica's hypothesis is valid. In consequence, factual universe may be quantum (non-classical) or not, but the union of factual and counter-factual universes is always classical.
Quantum non-locality has become a popular term. Yet, its precise meaning, and even its mere existence, is the subject of controversies. The main cause of the controversies is the never ending discussion on the appropriate definitions of Locality and Realism in the derivation of Bell's inequalities. On the other hand, Louis Sica derived Bell's inequalities from the hypothesis that time series of outcomes observed in one station do not change if the setting in the other (distant) station is changed. The derivation is based on arithmetical properties of time series only; it does not involve the definitions of Locality, Realism and probabilities, and is valid for series of any length. An important consequence is that violation of Bell's inequalities implies the series recorded (at the same time) in one station to be different if the setting in the other station is changed. This result gives precise meaning to the idea of quantum non-locality, and also makes evident why using it for faster than light signaling is impossible (because the difference is between factual and counter-factual series). Finally, it is demonstrated that series of outcomes, even if they violate Bell's inequalities, can be always embedded in a set of factual and counter-factual data that does hold to Bell's inequalities. Loosely speaking, the factual world may be quantum or not, but the union of factual and counter-factual worlds is always classical.
The problem of the existence of nonlocal effects in Quantum Mechanics is discussed. The problem is divided in two: the first ('soft') one is to explain the violation of Bell's inequalities as a statistical magnitude. This can be achieved by a simple model within non-Boolean Locality and Realism. This result shows that quantum non-Locality as a consequence of the statistical violation of Bell's inequalities is inexistent. The second ('hard') problem is to explain the violation as it is calculated from series of detection outcomes. L.Sica has demonstrated that, in order to violate Bell's inequalities, the series recorded at (say) Bob when the setting at station Alice is alfa, can be different from the series that would have been recorded at Bob if that setting had been alfa'instead. Therefore, non-Locality in the series of detection outcomes does exist. It cannot be experimentally verified because of its counterfactual nature, but is observed in computer simulations. An appropriate computer code is based on the simple model mentioned plus a contextual instruction. It explains 'how' (Sica's) non-Locality arises, and solves the hard problem. 'Why' the contextual instruction exists is explained by Hellwig and Kraus' postulate of covariant quantum state collapse. In consequence, (Sica's) non-Locality is not in contradiction with Relativity but, quite the opposite, it is implied by Relativistic covariance.
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.
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.
The interpretation of the meaning of Quantum Mechanics has faced controversy since its inception. Bell's inequalities are a touchstone in this controversy. Their observed violation demonstrates that at least one of the hypotheses involved in their derivation and test is false in Nature. In principle, one has to choose between accepting that Locality is false, what implies a possible contradiction with the Theory of Relativity, or accepting that Realism is false, what means to give up the existence of a physical world independent of the observer. The right answer has consequences both foundational and practical, and theoretical discussions have searched it for decades. We report the results of a Bell's experiment designed and performed to add observational information to the discussion. Three proposals to reveal the false hypothesis are carried out, namely: search of attractors in time series of observations, variation of randomness of binary series of outcomes between space-like and not-space-like separated conditions of observation, and test of a bound of Kolmogorov's complexity. The results are consistent with the absence of the specific form of Realism usually involved in the derivation of Bell's inequalities, while remaining compatible with Locality within the sensitivity of the tests. Independently of the foundational problem and of any interpretation, some bare observations have immediate practical impact on the best use of device-independent quantum Random Number Generators and Quantum Key Distribution.