We develop a hypergraph-theoretic framework for Spekkens contextuality applied to Kochen-Specker (KS) type scenarios that goes beyond the Cabello-Severini-Winter (CSW) framework. To do this, we add new hypergraph-theoretic ingredients to the CSW framework. We then obtain noise-robust noncontextuality inequalities in this generalized framework by applying the assumption of (Spekkens) noncontextuality to both preparations and measurements. The resulting framework goes beyond the CSW framework in both senses, conceptual and technical. On the conceptual level: 1) as in any treatment based on the generalized notion of noncontextuality à la Spekkens, we relax the assumption of outcome determinism inherent to the Kochen-Specker theorem but retain measurement noncontextuality, besides introducing preparation noncontextuality, 2) we do not require the exclusivity principle -- that pairwise exclusive measurement events must all be mutually exclusive -- as a fundamental constraint on measurement events of interest in an experimental test of contextuality, given that this property is not true of general quantum measurements, and 3) as a result, we do not need to presume that measurement events of interest are ``sharp" (for any definition of sharpness), where this notion of sharpness is meant to imply the exclusivity principle. On the technical level, we go beyond the CSW framework in the following senses: 1) we introduce a source events hypergraph -- besides the measurement events hypergraph usually considered -- and define a new operational quantity Corr that appears in our inequalities, 2) we define a new hypergraph invariant -- the weighted max-predictability -- that is necessary for our analysis and appears in our inequalities, and 3) our noise-robust noncontextuality inequalities quantify tradeoff relations between three operational quantities -- Corr , R , and p0 -- only one of which (namely, R ) corresponds to the Bell-Kochen-Specker functionals appearing in the CSW framework; when Corr=1 , the inequalities formally reduce to CSW type bounds on R . Along the way, we also consider in detail the scope of our framework vis-à-vis the CSW framework, particularly the role of Specker's principle in the CSW framework, i.e., what the principle means for an operational theory satisfying it and why we don't impose it in our framework.
We develop a hypergraph-theoretic framework for Spekkens contextuality applied to Kochen-Specker (KS) type scenarios that goes beyond the Cabello-Severini-Winter (CSW) framework. To do this, we add new hypergraph-theoretic ingredients to the CSW framework. We then obtain noise-robust noncontextuality inequalities in this generalized framework by applying the assumption of (Spekkens) noncontextuality to both preparations and measurements. The resulting framework goes beyond the CSW framework in both senses, conceptual and technical. On the conceptual level: 1) as in any treatment based on the generalized notion of noncontextuality à la Spekkens, we relax the assumption of outcome determinism inherent to the Kochen-Specker theorem but retain measurement noncontextuality, besides introducing preparation noncontextuality, 2) we do not require the exclusivity principle -- that pairwise exclusive measurement events must all be mutually exclusive -- as a fundamental constraint on measurement events of interest in an experimental test of contextuality, given that this property is not true of general quantum measurements, and 3) as a result, we do not need to presume that measurement events of interest are ``sharp" (for any definition of sharpness), where this notion of sharpness is meant to imply the exclusivity principle. On the technical level, we go beyond the CSW framework in the following senses: 1) we introduce a source events hypergraph -- besides the measurement events hypergraph usually considered -- and define a new operational quantity Corr that appears in our inequalities, 2) we define a new hypergraph invariant -- the weighted max-predictability -- that is necessary for our analysis and appears in our inequalities, and 3) our noise-robust noncontextuality inequalities quantify tradeoff relations between three operational quantities -- Corr , R , and p0 -- only one of which (namely, R ) corresponds to the Bell-Kochen-Specker functionals appearing in the CSW framework; when Corr=1 , the inequalities formally reduce to CSW type bounds on R . Along the way, we also consider in detail the scope of our framework vis-à-vis the CSW framework, particularly the role of Specker's principle in the CSW framework, i.e., what the principle means for an operational theory satisfying it and why we don't impose it in our framework.
Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framework to obtain noise-robust witnesses of contextuality from such scenarios. Our approach builds on the results of R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015), by providing new insights into the relationship between the structure of a contextuality scenario and the associated noise-robust noncontextuality inequalities that witness contextuality. The present work also forms a necessary counterpart to the framework presented in R. Kunjwal, Quantum 3, 184 (2019), which only applies to KS-colourable contextuality scenarios, i.e., those which do not admit logical proofs of the KS theorem but do admit statistical proofs. We rely on a single hypergraph invariant, defined in R. Kunjwal, Quantum 3, 184 (2019), that appears in our contextuality witnesses, namely, the weighted max-predictability. The present work can also be viewed as a study of this invariant. Significantly, unlike the case of R. Kunjwal, Quantum 3, 184 (2019), none of the graph invariants from the graph-theoretic framework for KS-contextuality due to Cabello, Severini, and Winter (the ``CSW framework", Phys. Rev. Lett. 112, 040401 (2014)) are relevant for our noise-robust noncontextuality inequalities.
A characterization of noncontextual models which fall within the ambit of Fine's theorem is provided. In particular, the equivalence between the existence of three notions is made explicit: a joint probability distribution over the outcomes of all the measurements considered, a measurement-noncontextual and outcome-deterministic (or KS-noncontextual, where KS stands for Kochen-Specker) model for these measurements, and a measurement-noncontextual and factorizable model for them. A KS inequality, therefore, is implied by each of these three notions. Following this characterization of noncontextual models that fall within the ambit of Fine's theorem, nonfactorizable noncontextual models which lie outside the domain of Fine's theorem are considered. While outcome determinism for projective (sharp) measurements in quantum theory can be shown to follow from the assumption of preparation noncontextuality, such a justification is not available for nonprojective (unsharp) measurements which ought to admit outcome-indeterministic response functions. The Liang-Spekkens-Wiseman (LSW) inequality is cited as an example of a noncontextuality inequality that should hold in any noncontextual model of quantum theory without assuming factorizability. Three other noncontextuality inequalities, which turn out to be equivalent to the LSW inequality under relabelings of measurement outcomes, are derived for Specker's scenario. The polytope of correlations admissible in this scenario, given the no-disturbance condition, is characterized.
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.
So far, most of the literature on (quantum) contextuality and the Kochen–\nSpecker theorem seems either to concern particular examples of contextuality, or be\nconsidered as quantum logic. Here, we develop a general formalism for contextuality\nscenarios based on the combinatorics of hypergraphs, which significantly refines a similar\nrecent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we\nexplicitly include the normalization of probabilities, which gives us a much finer control\nover the various sets of probabilistic models like classical, quantum and generalized\nprobabilistic. In particular, our framework specializes to (quantum) nonlocality in the\ncase of Bell scenarios, which arise very naturally from a certain product of contextuality\nscenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to\nseveral graph invariants. The recently proposed Local Orthogonality principle turns out\nto be a special case of a general principle for contextuality scenarios related to the Shannon\ncapacity of graphs. Our results imply that it is strictly dominated by a low level of\nthe Navascués–Pironio–Acín hierarchy of semidefinite programs, which we also apply\nto contextuality scenarios.\nWe derive a wealth of results in our framework, many of these relating to quantum\nand supraquantum contextuality and nonlocality, and state numerous open problems.\nFor example, we show that the set of quantum models on a con
When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen–Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen–Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres–Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres–Mermin square.
A new theory-independent noncontextuality inequality is presented [R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015)] based on a Kochen-Specker (KS) set without imposing the assumption of determinism. By proposing noncontextuality inequalities, we show that such result can be generalized from a KS set to the noncontextuality inequalities not only for a state-independent scenario but also for a state-dependent scenario. The YO-13 ray and n cycle ray are considered as examples.