Locality and non-contextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic no-go theorems by Bell, Kochen-Specker, et al. is to show that non-locality and contextuality are necessary features of any theory whose predictions agree with those of quantum mechanics. We use the mathematics of sheaf theory to analyze the structure of non-locality and contextuality in a very general setting. Starting from a simple experimental scenario, and the kind of probabilistic models familiar from discussions of Bell’s theorem, we show that there is a very direct, compelling formalization of these notions in sheaf-theoretic terms. Moreover, on the basis of this formulation, we show that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. We give linear algebraic methods for computing these obstructions, and use these methods to obtain a number of new insights into non-locality and contextuality. For example, we distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples — due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively — occupy successively higher levels of this hierarchy. We show how our abstract setting can be represented in quantum mechanics. In doing so, we uncover a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.
Locality and non-contextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic no-go theorems by Bell, Kochen-Specker, et al. is to show that non-locality and contextuality are necessary features of any theory whose predictions agree with those of quantum mechanics. We use the mathematics of sheaf theory to analyze the structure of non-locality and contextuality in a very general setting. Starting from a simple experimental scenario, and the kind of probabilistic models familiar from discussions of Bell’s theorem, we show that there is a very direct, compelling formalization of these notions in sheaf-theoretic terms. Moreover, on the basis of this formulation, we show that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. We give linear algebraic methods for computing these obstructions, and use these methods to obtain a number of new insights into non-locality and contextuality. For example, we distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples — due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively — occupy successively higher levels of this hierarchy. We show how our abstract setting can be represented in quantum mechanics. In doing so, we uncover a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.
Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,l) and (2,k,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.
In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.
Since Bell, Kochen and Specker formulated the first notions of contextuality, a number of frameworks have been created to help make philosophical and mathematical sense of the concept. Two of the most common are the Equivalence-based approach due to Spekkens, and the Sheaf-theoretic framework of Abramsky and Brandenburger. Each of these can support a hierarchy of strengths of contextuality, which can include possibilistic concepts. This thesis will explore the differences between the two approaches, and explore what can be demonstrated by using weaker notions of noncontextuality. Notions of noncontextuality are important for proving no-go theorems which highlight that many properties that are true of classical mechanics fail to be compatible with quantum predictions. Often, the notions of noncontextuality used are very strong assumptions, and even weaker flavours of noncontextuality can be used to draw the same, or similar conclusions. An important class of such notions are, possibilistic assumptions that only care about whether or not events are possible or impossible, rather than the exact Born rule probabilities that are applied to events. Even these weaker notions are often impossible to resolve with the logical structure and predictions of quantum theory. Contextuality has been demonstrated to be a resource for many quantum computational operations, and the contextual fraction has been shown to be an important resource. We introduce a metric that can differentiate between different quantum mechanical scenarios with contextual fraction 1, and demonstrate analogues of CHSH and Tsirelson bounds for this quantity. We also investigate the computational complexity of calculating the values of some metrics for nonlocality and contextuality, completing the classification of possibilistic nonlocality scenarios, as well as other metrics of nonlocality analogous to our metric of maximal contextuality.
A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.
The sheaf-theoretic structure is useful in classifying no-go theorems related to non-locality and contextuality. It provides a new point of view different from conventional formularization of quantum mechanics. First, we examine a relationship between the conventional formularization and the innovative formularization. There exists an equivalence of their categories, and from the equivalence, one locality can be transformed to another as a concrete example. Next, we extend the quantum mechanics which has a finite-degree of freedom to the quantum filed theory with an infinite-degree of freedom, especially to the algebraic quantum field theory (AQFT for short). We consider about a violation of the Bell inequality in AQFT, and we show that the condition of strict spacelike separation has the same Cartesian product structure as locality of quantum mechanics. Also, we show that no-signalling property can be proved by Split Property. A local state is a sheaf which is defined by Split Property in AQFT. It induces the sheaf-theoretic structure. Finally, we show an extension of the No-Signalling theorem which depends on spacetime in AQFT.