We address the challenge of extending the criterion for Kochen-Specker contextuality to data with disturbance, to accommodate experimental imperfections or applications to macroscopic settings beyond quantum physics. We find that any such extension must abandon one or more principles that are core to the concept of contextuality. Specifically, we prove that no extended criterion agreeing with Kochen-Specker contextuality in the absence of disturbance can simultaneously satisfy the following: (1) any deterministic system is noncontextual; (2) discarding information cannot turn a noncontextual system into a contextual one; (3) classical post-processing cannot create contextuality; and (4) the joint realization of two statistically independent noncontextual systems is noncontextual. From the standpoint of resource theories, the obstruction arises because free operations can turn noncontextual disturbing behaviors into contextual non-disturbing ones. We suggest that future work on extended contextuality needs to move beyond the standard model-independent framework and account for the causal structure responsible for disturbance.
We address the challenge of extending the criterion for Kochen-Specker contextuality to data with disturbance, to accommodate experimental imperfections or applications to macroscopic settings beyond quantum physics. We find that any such extension must abandon one or more principles that are core to the concept of contextuality. Specifically, we prove that no extended criterion agreeing with Kochen-Specker contextuality in the absence of disturbance can simultaneously satisfy the following: (1) any deterministic system is noncontextual; (2) discarding information cannot turn a noncontextual system into a contextual one; (3) classical post-processing cannot create contextuality; and (4) the joint realization of two statistically independent noncontextual systems is noncontextual. From the standpoint of resource theories, the obstruction arises because free operations can turn noncontextual disturbing behaviors into contextual non-disturbing ones. We suggest that future work on extended contextuality needs to move beyond the standard model-independent framework and account for the causal structure responsible for disturbance.
When it isn't possible to tell two distinct experimental procedures apart purely from their input/output statistics, then it seems a plausible hypothesis that the two procedures must be physically identical. We call such a hypothesis "noncontextuality", an instance of Leibniz's principle of the identity of indiscernibles. Read in the contrapositive, this hypothesis entails that any physical distinctions between two experimental procedures must necessarily lead to a difference in their operational statistics. The results I present in this thesis concern the failure of this hypothesis -- a failure dubbed "contextuality" -- when one tries to embed an operational theory (such as quantum theory) in the ontological models framework. The Kochen-Specker theorem demonstrates the failure of noncontextuality for deterministic ontological models of quantum theory, i.e., those ontological models where the ontic/physical state of the system fixes the outcome of any projective measurement on the system in a deterministic manner. This thesis goes beyond the Kochen-Specker (KS) theorem by asking what operational facts must be verified in experiments to conclude that Nature does not admit noncontextual ontological models, not even indeterministic ones. This leads to noncontextuality inequalities that are robust to noise in the preparations and measurements. In the particular case of quantum theory, these inequalities are meaningful even when unsharp measurements (or POVMs) are allowed, a feature not shared by the traditional approach to KS-noncontextuality where unsharp measurements are excluded by fiat: allowing them renders even trivial POVMs (proportional to identity) maximally KS-contextual. The sense in which trivial POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they are simply too noisy to lead to violation of our noncontextuality inequalities.
A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here, compatible measurements are those that can be performed simultaneously or in any order without disturbance. This conflict is generically called quantum contextuality. In this article, we present an introduction to this subject and its current status. We review several proofs of the Kochen-Specker theorem and different notions of contextuality. We explain how to experimentally test some of these notions and discuss connections between contextuality and nonlocality or graph theory. Finally, we review some applications of contextuality in quantum information processing.
The question of quantum contextuality in the Mermin‐Peres square is considered. It is shown that a deterministic, noncontextual hidden variable model of this problem is not inconsistent with quantum mechanics, contrary to the Kochen‐Specker theorem. The key idea is that measurement outcomes may be viewed as deterministic functions of hidden variable states which are disturbed through the process of measurement. This, in turn, implies that the outcome of measuring the product of two commensurate observables need not be equal to the product of the outcomes that would have been obtained had they been measured individually. A critical analysis of some recent and proposed experimental tests of contextuality is also provided.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the structural essence of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often remains unclear. This issue already looms large at the level of the Bell-Kochen-Specker theorem: while traditional proofs proceed by showing the nonexistence of valuations, the notion of state-independent contextuality in the marginal approach allows to prove contextuality from seemingly weaker assumptions. In the light of this, and at the absence of a unified mathematical framework for Kochen-Specker contextuality, the original algebraic approach has been widely abandoned, in favour of the study of contextual correlations. Here, we reinstate the algebraic perspective on contextuality. Concretely, by building on the novel concept of context connections, we reformulate the algebraic relations between observables originally postulated by Kochen and Specker, and we explicitly demonstrate their consistency with the notion of state-independent contextuality. In the present paper, we focus on the new conceptual ideas and discuss them in the concrete setting of spin-1 observables, specifically those in the example of [S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012)]; in a companion paper, we generalise these ideas, obtain a complete characterisation of Kochen-Specker contextuality and provide a detailed comparison with the related notions of contextuality in the marginal and graph-theoretic approach.
If noncontextuality is defined as the robustness of a system's response to a measurement against other simultaneous measurements, then the Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality. Namely, for the argument to be effective, (i) each operator must be uniquely associated with a measurement and (ii) commuting operators must represent simultaneous measurements. However, in all Kochen-Specker arguments discussed in the literature either (i) or (ii) is not met. Arguments meeting (i) contain at least one subset of mutually commuting operators which do not represent simultaneous measurements and hence fail to physically justify the functional composition principle. Arguments meeting (ii) associate some operators with more than one measurement and hence need to invoke an extra assumption different from noncontextuality.
The view that non-classical phenomena are witnessed when noncontextuality inequalities are violated by quantum correlations has become widespread over the last decades, and many authors describe contextuality as a counter-intuitive and non-classical feature of quantum systems. In this paper we show that a classical input/output device generating contextual data (i.e., data violating noncontextuality inequalities) can easily be found if the following conditions are satisfied: (1) We only have access to a specific collection of epistemic measurements, namely measurements which do not necessarily reveal properties possessed by the system to the agent, and (2) not all these measurements can be jointly performed. The conclusion we draw from this example is twofold. Firstly, regarding model-independent contextuality, it suggests that, unlike Bell inequalities (whose violation allegedly implies non-locality), noncontextuality inequalities have no phenomenological significance per se: we cannot treat experiments as black-boxes and draw conclusions about the phenomenological status of their underlying structure based only on contextuality analysis of the data they generate. Secondly, regarding quantum theory, it indicates that one who accepts, as many interpretations suggest, that quantum theory satisfies both these conditions should not take for granted that quantum contextuality is a signature of non-classicality.