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Camillo G, Cervantes VH. Measures of contextuality in cyclic systems and the negative probabilities measure CNT 3. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2024; 382:20230007. [PMID: 38281723 DOI: 10.1098/rsta.2023.0007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Accepted: 08/21/2023] [Indexed: 01/30/2024]
Abstract
Several principled measures of contextuality have been proposed for general systems of random variables (i.e. inconsistently connected systems). One such measure is based on quasi-couplings using negative probabilities (here denoted by [Formula: see text], Dzhafarov & Kujala, 2016 Quantum interaction). Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377, 20190149. (doi:10.1098/rsta.2019.0149)) introduced a measure of contextuality, [Formula: see text], that naturally generalizes to a measure of non-contextuality. Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377, 20190149. (doi:10.1098/rsta.2019.0149)) additionally conjectured that in the class of cyclic systems these two measures are proportional. Here we prove that conjecture is correct. Recently, Cervantes (Cervantes 2023 J. Math. Psychol. 112, 102726. (doi:10.1016/j.jmp.2022.102726)) showed the proportionality of [Formula: see text] and the Contextual Fraction measure introduced by Abramsky & Brandenburger (Abramsky & Brandenburger 2011 New J. Phys. 13, 113036. (doi:10.1088/1367-2630/13/11/113036)). The present proof completes the description of the interrelations of all contextuality measures proposed within or translated into the Contextuality-by-Default framework so far as they pertain to cyclic systems. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
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Affiliation(s)
| | - Víctor H Cervantes
- Instituto de Física, Universidade de São Paulo, Brasil
- Department of Psychology, University of Illinois Urbana, Champaign, IL, USA
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Amaral B, Terra Cunha M. On geometrical aspects of the graph approach to contextuality. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2024; 382:20230008. [PMID: 38281724 DOI: 10.1098/rsta.2023.0008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 12/23/2023] [Indexed: 01/30/2024]
Abstract
The connection between contextuality and graph theory has paved the way for numerous advancements in the field. One notable development is the realization that sets of probability distributions in many contextuality scenarios can be effectively described using well-established convex sets from graph theory. This geometric approach allows for a beautiful characterization of these sets. The application of geometry is not limited to the description of contextuality sets alone; it also plays a crucial role in defining contextuality quantifiers based on geometric distances. These quantifiers are particularly significant in the context of the resource theory of contextuality, which emerged following the recognition of contextuality as a valuable resource for quantum computation. In this paper, we provide a comprehensive review of the geometric aspects of contextuality. Additionally, we use this geometry to define several quantifiers, offering the advantage of applicability to other approaches to contextuality where previously defined quantifiers may not be suitable. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
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Affiliation(s)
- Barbara Amaral
- Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil
- Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brazil
- Departamento de Física e Matemática, CAP - Universidade Federal de São João del-Rei, 36.420-000, Ouro Branco, Minas Gerais, Brazil
- International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, PO Box 1613, Natal, Brazil
- Department of Mathematical Physics, Institute of Physics, University of São Paulo, R. do Matão 1371, São Paulo 05508-090, Brazil
| | - Marcelo Terra Cunha
- Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil
- Departamento de Matemática Aplicada, IMECC-Unicamp, 13084-970, Campinas, São Paulo, Brazil
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Vallée K, Emeriau PE, Bourdoncle B, Sohbi A, Mansfield S, Markham D. Corrected Bell and non-contextuality inequalities for realistic experiments. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2024; 382:20230011. [PMID: 38281716 DOI: 10.1098/rsta.2023.0011] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 10/20/2023] [Indexed: 01/30/2024]
Abstract
Contextuality is a feature of quantum correlations. It is crucial from a foundational perspective as a non-classical phenomenon, and from an applied perspective as a resource for quantum advantage. It is commonly defined in terms of hidden variables, for which it forces a contradiction with the assumptions of parameter-independence and determinism. The former can be justified by the empirical property of non-signalling or non-disturbance, and the latter by the empirical property of measurement sharpness. However, in realistic experiments neither empirical property holds exactly, which leads to possible objections to contextuality as a form of non-classicality, and potential vulnerabilities for supposed quantum advantages. We introduce measures to quantify both properties, and introduce quantified relaxations of the corresponding assumptions. We prove the continuity of a known measure of contextuality, the contextual fraction, which ensures its robustness to noise. We then bound the extent to which these relaxations can account for contextuality, via corrections terms to the contextual fraction (or to any non-contextuality inequality), culminating in a notion of genuine contextuality, which is robust to experimental imperfections. We then show that our result is general enough to apply or relate to a variety of established results and experimental set-ups. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
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Affiliation(s)
- Kim Vallée
- Sorbonne Université, CNRS, LIP6, Paris 75005, France
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Kupczynski M. Contextuality-by-Default Description of Bell Tests: Contextuality as the Rule and Not as an Exception. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1104. [PMID: 34573729 PMCID: PMC8468053 DOI: 10.3390/e23091104] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/25/2021] [Revised: 08/13/2021] [Accepted: 08/20/2021] [Indexed: 11/16/2022]
Abstract
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.
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Affiliation(s)
- Marian Kupczynski
- Département d'informatique et d'ingénierie, Université du Québec en Outaouais, Case Postale 1250, Succursale Hull, Gatineau, QC J8X 3X7, Canada
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Cervantes VH, Dzhafarov EN. Contextuality Analysis of Impossible Figures. ENTROPY 2020; 22:e22090981. [PMID: 33286751 PMCID: PMC7597294 DOI: 10.3390/e22090981] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/05/2020] [Revised: 08/31/2020] [Accepted: 08/31/2020] [Indexed: 11/30/2022]
Abstract
This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense.
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Affiliation(s)
- Víctor H. Cervantes
- Department of Psychology, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA;
| | - Ehtibar N. Dzhafarov
- Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USA
- Correspondence:
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Indistinguishability and Negative Probabilities. ENTROPY 2020; 22:e22080829. [PMID: 33286600 PMCID: PMC7517418 DOI: 10.3390/e22080829] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/19/2020] [Revised: 07/27/2020] [Accepted: 07/28/2020] [Indexed: 11/17/2022]
Abstract
In this paper, we examined the connection between quantum systems' indistinguishability and signed (or negative) probabilities. We do so by first introducing a measure-theoretic definition of signed probabilities inspired by research in quantum contextuality. We then argue that ontological indistinguishability leads to the no-signaling condition and negative probabilities.
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Dzhafarov EN. On joint distributions, counterfactual values and hidden variables in understanding contextuality. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190144. [PMID: 31522638 DOI: 10.1098/rsta.2019.0144] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 05/13/2019] [Indexed: 06/10/2023]
Abstract
This paper deals with three traditional ways of defining contextuality: (C1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts; (C2) in terms of relationship between factual measurements in a given context and counterfactual measurements that could be made if one used other contexts; and (C3) in terms of (non)existence of 'hidden variables' that determine the outcomes of all factually performed measurements. It is generally believed that the three meanings are equivalent, but the issues involved are not entirely transparent. Thus, arguments have been offered that C2 may have nothing to do with C1, and the traditional formulation of C1 itself encounters difficulties when measurement outcomes in a contextual system are treated as random variables. I show that if C1 is formulated within the framework of the Contextuality-by-Default (CbD) theory, the notion of a probabilistic coupling, the core mathematical tool of CbD, subsumes both counterfactual values and 'hidden variables'. In the latter case, a coupling itself can be viewed as a maximally parsimonious choice of a hidden variable. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
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