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arXiv:2108.05480 · Entropy

Assumption-Free Derivation of the Bell-Type Criteria of Contextuality/Nonlocality

Ehtibar N. Dzhafarov

Published Nov 19, 2021• 9 citations
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Abstract

Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of contextuality in the contextuality-by-default approach, and then specializing these criteria to consistently connected systems.

DOI
10.3390/e23111543

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Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of contextuality in the contextuality-by-default approach, and then specializing these criteria to consistently connected systems.
Traditional Bell criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell criteria for systems with disturbance, based on the generalized definition of contextuality in the Contextuality-by-Default approach, and then specializing these criteria to systems with no disturbance. Context-independent mapping and no-fine-tuning are derived rather than assumed, and free choice follows from context-independent mapping. KEYWORDS: Bell criteria; contextuality; context-independent mapping; free choice; local causality; no-fine-tuning; nonlocality.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality, on the other hand, is on a more controversial foundation. In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality. First, I show that Bell inequalities can be derived solely from the assumptions of no-signalling and no-fine-tuning of the causal model. This removes two extra assumptions from a recent result from Wood and Spekkens, and remarkably, does not require any assumption related to independence of measurement settings -- unlike all other derivations of Bell inequalities. I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning. Thus the quantum violation of classical causality goes beyond the case of space-like separated systems, and manifests already in scenarios involving single systems.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks.Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure.Contextuality, on the other hand, is on a more controversial foundation.In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality.First, I show that Bell inequalities can be derived solely from the assumptions of no signaling and no fine-tuning of the causal model.This removes two extra assumptions from a recent result from Wood and Spekkens and, remarkably, does not require any assumption related to independence of measurement settings-unlike all other derivations of Bell inequalities.I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning.Thus, the quantum violation of classical causality goes beyond the case of spacelike-separated systems and already manifests in scenarios involving single systems.
Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems' couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory T is contextually equivalent to a theory T' in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in T and T' such that the corresponding systems in T and T' are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together.
Abstract Free choice (or statistical independence) assumption in a hidden variable model (HVM) means that the settings chosen by experimenters do not depend on the values of the hidden variable. The assumption of context-independent (CI) mapping in an HVM means that the results of a measurement do not depend on settings for other measurements. If the measurements are spacelike separated, this assumption is known as local causality. Both free choice and CI mapping assumptions are considered necessary for derivation of the Bell-type criteria of contextuality/nonlocality. It is known, however, for a variety of special cases, that the two assumptions are not logically independent. We show here, in complete generality, for any system of random variables with or without disturbance/signaling, that an HVM that postulates CI mapping is equivalent to an HVM that postulates free choice. If one denies the possibility that a given empirical scenario can be described by an HVM in which measurements depend on other measurements’ settings, free choice violations should be denied too, and vice versa.
Contextuality and entanglement are valuable resources for quantum computing and quantum information. Bell inequalities are used to certify entanglement; thus, it is important to understand why and how they are violated. Quantum mechanics and behavioural sciences teach us that random variables 'measuring' the same content (the answer to the same Yes or No question) may vary, if 'measured' jointly with other random variables. Alice's and BoB's raw data confirm Einsteinian non-signaling, but setting dependent experimental protocols are used to create samples of coupled pairs of distant ±1 outcomes and to estimate correlations. Marginal expectations, estimated using these final samples, depend on distant settings. Therefore, a system of random variables 'measured' in Bell tests is inconsistently connected and it should be analyzed using a Contextuality-by-Default approach, what is done for the first time in this paper. The violation of Bell inequalities and inconsistent connectedness may be explained using a contextual locally causal probabilistic model in which setting dependent variables describing measuring instruments are correctly incorporated. We prove that this model does not restrict experimenters' freedom of choice which is a prerequisite of science. Contextuality seems to be the rule and not an exception; thus, it should be carefully tested.