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.
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.
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.
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.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.
Strong O-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ({0,1}-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the {0,1} case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset O⊂[0,1]. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of O that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong O-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.
Since being defined by Kochen and Specker, and separately by Bell, contextuality has been proposed as one of the key phenomena that distinguishing quantum theory from classical theories. However, with the rise of quantum information contextuality's position as the leading definition of the quantum/classical boundary has been called into question. This is due to the fact that a contextual explanation is required by subtheories that offer no exponential quantum computational advantage over classical computation. In this thesis, submitted in requirement for a PhD in physics with quantum information, we shall explore this unwanted contextuality, and show that generalized contextuality is more prevalent than was previously thought. \n \nFirst we will show that the single-qubit stabilizer subtheory, which was previously thought of as a non-contextual subtheory, requires a generalized contextual ontological model, when transformations are included in the operational description. In addition to this we show that the even smaller single-rebit stabilizer subtheory, a strict subtheory of the single-qubit stabilizer subtheory, must also admit a generalized contextual ontological model, again when transformations are included in the operational description. We then show that both these subtheories require negatively represented quasi-probability representations, re-establishing the link between contextuality and negativity for transformations. We also investigate the representation of transformations in generic quasi-probability representations, showing that under a reasonable assumption almost-all unitaries must be negatively represented by a finite quasi-probability representation. \n \nSecond we will investigate the efficiently simulable n-qubit stabilizer subtheory, which exhibits all forms of contextuality and thus represents the main obstacle to identifying contextuality as a resource for quantum computation. To this end we present an attempt at constructing a model based on a frame-switching Wigner function. This leads us to constructing a contextual ψ-epistemic ontological model of the n-qubit stabilizer formalism. We shall see that this model is outcome deterministic, which is one of the core assumptions in the definition of traditional non-contextuality. This model then will lead us to a result that proves that any ontological model of the n-qubit stabilizer formalism requires at least n-1 generators to be encoded in the ontology of the model. As n-1 generators represents almost full knowledge about the stabilizer state, we therefore posit that ψ-onicity, a more controversial notion of non-classicality, is actually the resource for quantum computation.
It is well known that in quantum mechanics we cannot always define consistently properties that are context independent.Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities.In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability.In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen-Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same.We will discuss how the consequences of this move may help us understand quantum contextuality.