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arXiv:2310.19383 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Corrected Bell and non-contextuality inequalities for realistic experiments

Kim Vallée, Pierre-Emmanuel Emeriau, Boris Bourdoncle, Adel Sohbi, Shane Mansfield, Damian Markham

Published Jan 28, 2024• 6 citations
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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'.

DOI
10.1098/rsta.2023.0011

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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'.
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.
Quantum non-locality and quantum contextuality result from the impossibility of explain- ing the probabilities of measurement outcomes in quantum systems using theoretical mod- els that assume that observables have well-defined values prior to the measurement, and that these values do not depend on which other compatible measurements are performed simultaneously to the measurement. These two features are not only of foundational im- portance for quantum theory, but are also recognized as resources for quantum computing and other applications. The experimental verification of contextuality has garnered significant attention, and its recognition as an operational resource for quantum information is expected to further fuel this interest. Developing a robust theoretical framework for contextuality that is both operationally grounded and experimentally applicable is crucial. Particularly, real-world experiments often violate the non-disturbance assumption, requiring alternative frame- works that remain effective under such conditions. This dissertation builds upon a recent work by Renou et.al, which explores how dif- ferent variants of a Bell inequality, despite being equivalent under the condition of no- signaling, are not equally robust in real experiments. In order to do so, they propose a method which employs the group of symmetries of a Bell scenario to analyze and op- timize the robustness of Bell inequalities. The primary goal of this dissertation is to generalize this method to contextuality scenarios, exploring the role of group representa- tion theory in optimizing inequalities, and enhancing both theoretical understanding and experimental applicability.
The operational characterization of contextuality is the cornerstone in the development of resource theory of contextuality. Here, we introduce contextual robustness as an operational measure of contextuality on empirical models. We first show that it satisfies all the properties a proper contextual measure should satisfy. Besides, we derive the linear programing of contextual robustness and prove that contextual robustness exactly characterizes the maximal violation of certain types of Bell inequalities. Moreover, we show that contextual robustness quantifies the accessible advantage of empiric models in measurement-based quantum computation, and provide a tighter upper bound on accessible advantage in certain cases.
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.
We consider the contextual fraction as a quantitative measure of contextuality of empirical models, i.e., tables of probabilities of measurement outcomes in an experimental scenario. It provides a general way to compare the degree of contextuality across measurement scenarios; it bears a precise relationship to violations of Bell inequalities; its value, and a witnessing inequality, can be computed using linear programing; it is monotonic with respect to the "free" operations of a resource theory for contextuality; and it measures quantifiable advantages in informatic tasks, such as games and a form of measurement-based quantum computing.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.