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.
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.
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.
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'.
Motivated by the importance of contextuality and a work on the robustness of the entanglement of mixed quantum states, the robustness of contextuality (RoC) R C ( e ) of an empirical model e against non-contextual noises was introduced and discussed in Science China Physics, Mechanics and Astronomy (59(4) and 59(9), 2016). Because noises are not always non-contextual, this paper introduces and discusses the generalized robustness of contextuality (GRoC) R g ( e ) of an empirical model e against general noises. It is proven that R g ( e ) = 0 if and only if e is non-contextual. This means that the quantity R g can be used to distinguish contextual empirical models from non-contextual ones. It is also shown that the function R g is convex on the set of all empirical models and continuous on the set of all no-signaling empirical models. For any two empirical models e and f such that the generalized relative robustness of e with respect to f is finite, a fascinating relationship between the GRoCs of e and f is proven, which reads R g ( e ) R g ( f ) ≤ 1 . Lastly, for any n-cycle contextual box e, a relationship between the GRoC R g ( e ) and the extent Δ e of violating the non-contextual inequalities is established.
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.
Contextuality is a fundamental feature of quantum theory necessary for certain models of quantum computation and communication. Serious steps have therefore been taken towards a formal framework for contextuality as an operational resource. However, the main ingredient of a resource theory-a concrete, explicit form of free operations of contextuality-was still missing. Here we provide such a component by introducing noncontextual wirings: a class of contextuality-free operations with a clear operational interpretation and a friendly parametrization. We characterize them completely for general black-box measurement devices with arbitrarily many inputs and outputs. As applications, we show that the relative entropy of contextuality is a contextuality monotone and that maximally contextual boxes that serve as contextuality bits exist for a broad class of scenarios. Our results complete a unified resource-theoretic framework for contextuality and Bell nonlocality.
Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine--Abramsky--Brandenburger theorem extends to this setting, an important consequence of which is that nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, can also be defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to dis cretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program, and calculated with increasing accuracy using semi-definite programming approximations.