We introduce a framework to describe probabilistic models in Bell experiments, and more generally in contextuality scenarios. Such a scenario is a hypergraph whose vertices represent elementary events and hyperedges correspond to measurements. A probabilistic model on such a scenario associates to each event a probability, in such a way that events in a given measurement have a total probability equal to one. We discuss the advantages of this framework, like the unification of the notions of contexuality and nonlocality, and give a short overview of results obtained elsewhere.
We introduce a framework to describe probabilistic models in Bell experiments, and more generally in contextuality scenarios. Such a scenario is a hypergraph whose vertices represent elementary events and hyperedges correspond to measurements. A probabilistic model on such a scenario associates to each event a probability, in such a way that events in a given measurement have a total probability equal to one. We discuss the advantages of this framework, like the unification of the notions of contexuality and nonlocality, and give a short overview of results obtained elsewhere.
Quantum mechanics allows only certain sets of experimental results (or ``probabilistic models'') for Bell-type quantum nonlocality experiments. A derivation of this set from simple physical or information theoretic principles would represent an important step forward in our understanding of quantum mechanics, and this problem has been intensely investigated in recent years. ``Macroscopic locality,'' which requires the recovery of locality in the limit of large numbers of trials, is one of several principles discussed in the literature that place a bound on the set of quantum probabilistic models. A similar question can also be asked about probabilistic models for the more general class of quantum contextuality experiments. Here, we extend the macroscopic locality principle to this more general setting, using the hypergraph approach of Ac\'{\i}n, Fritz, Leverrier, and Sainz [Comm. Math. Phys. 334(2), 533--628 (2015)], which provides a framework to study both phenomena of nonlocality and contextuality in a unified manner. We find that the set of probabilistic models allowed by our macroscopic noncontextuality principle is equivalent to an important and previously studied set in this formalism, which is slightly larger than the quantum set. In the particular case of Bell scenarios, this set is equivalent to the set of ``almost-quantum'' models, which is of particular interest since the latter was recently shown to satisfy all but one of the principles that have been proposed to bound quantum probabilistic models, without being implied by any of them (or even their conjunction). Our condition is the first characterization of the almost-quantum set from a simple physical principle.
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
We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.
We introduce an operator system, universal for the probabilistic models of a contextuality scenario, and identify its maximal C∗-cover via the right C∗-algebra of a canonical ternary ring of operators, arising from a hypergraph version of stochastic operator matrices. We study dilating contextuality scenarios, which have the property that each positive operator representation thereof admits a dilation to a projective representation on a larger Hilbert space, and characterise them via the equality of the aforementioned universal operator system and the operator system arising from the canonical generators of the respective hypergraph C∗-algebra. We characterise the no-signalling probabilistic models over a pair of contextuality scenarios of different types, which arise from either the positive operator representations or from the projective representations of these scenarios, in terms of states on operator system tensor products. Generalising the notion of a synchronous no-signalling correlation to the hypergraph framework, we define coherent probabilistic models associated with a given contextuality scenario and characterise various classes thereof via different types of traces of the hypergraph C∗-algebra, associated with the scenario. We establish several equivalent formulations of the Connes Embedding Problem in terms of no-signalling probabilistic models and hypergraph operator systems.
Quantum theory departs from classical probabilistic theories in foundational ways. These departures--termed quantumness here--power quantum information and computation. This thesis charts the role of discrete structures in assessing quantumness, synthesizing elements of my postdoctoral research through this lens. After an introduction to the necessary background concepts, I present my work under three broad categories. First, I present work on contextuality that extensively relies on (undirected) graphs and hypergraphs as the discrete structures of interest; more specifically, it relies on invariants associated with them. This work includes Kochen-Specker (KS) contextuality and its operationalization to generalized contextuality, expressed via (hyper)graph-theoretic frameworks. I also present work on KS-contextuality in multiqubit systems and an application of generalized contextuality to a one-shot communication task, both of which rely on hypergraphs. Second, I present work on causality, where the discrete structures of interest are directed graphs. This includes work on indefinite causal order, specifically its connections to the gap between local operations and classical communication (LOCC) and separable operations (SEP), and a device-independent notion of nonclassicality--termed antinomicity--that generalizes Bell nonlocality without global causal assumptions. Finally, I present work on the incompatibility of quantum measurements, its connection to Bell nonlocality, and its role in discriminating between quantum and almost quantum correlations in the single-system setting. The discrete structures of interest here are hypergraphs that model joint measurability relations between quantum measurements. I conclude with a summary and an overview of work that is not covered in this thesis.
The presence of contextuality in quantum theory was first highlighted by Bell, Kochen and Specker, who discovered that for quantum systems of three or more dimensions, measurements could not be viewed as deterministically revealing pre-existing properties of the system. More precisely, no model can assign deterministic outcomes to the projectors of a quantum measurement in a way that depends only on the projector and not the context (the full set of projectors) in which it appeared, despite the fact that the Born rule probabilities associated with projectors are independent of the context. A more general, operational definition of contextuality introduced by Spekkens, which we will term 'probabilistic contextuality', drops the assumption of determinism and allows for operations other than measurements to be considered contextual. Even two-dimensional quantum mechanics can be shown to be contextual under this generalised notion. Probabilistic noncontextuality represents the postulate that elements of an operational theory that cannot be distinguished from each other based on the statistics of arbitrarily many repeated experiments (they give rise to the same operational probabilities) are ontologically identical. In this paper, we introduce a framework that enables us to distinguish between different noncontextuality assumptions in terms of the relationships between the ontological representations of objects in the theory given a certain relation between their operational representations. This framework can be used to motivate and define a 'possibilistic' analogue, encapsulating the idea that elements of an operational theory that cannot be unambiguously distinguished operationally can also not be unambiguously distinguished ontologically. We then prove that possibilistic noncontextuality is equivalent to an alternative notion of noncontextuality proposed by Hardy. Finally, we demonstrate that these weaker noncontextuality assumptions are sufficient to prove alternative versions of known 'no-go' theorems that constrain ψ -epistemic models for quantum mechanics.