Abstract 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 contex-tuality 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.
Abstract 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 contex-tuality 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.
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.
The Kochen–Specker (KS) theorem is a cornerstone result in quantum foundations, establishing that quantum correlations in Hilbert spaces of dimension d ≥ 3 cannot be explained by (consistent) hidden variable theories that assign a single deterministic outcome to each measurement. Specifically, there exist finite sets of vectors in these dimensions such that no non-contextual deterministic ({0, 1}) outcome assignment is possible obeying the rules of exclusivity and completeness—that the sum of assignments to every set of mutually orthogonal vectors be ≤1 and the sum of value assignments to any d mutually orthogonal vectors be equal to 1. Another central result in quantum foundations is Gleason’s theorem that justifies the Born rule as a mathematical consequence of the quantum formalism. The KS theorem can be seen as a consequence of Gleason’s theorem and the logical compactness theorem. In a similar vein, Gleason’s theorem also indicates the existence of KS-type finite vector constructions to rule out other finite-alphabet outcome assignments beyond the {0, 1} case. Here, we propose a generalisation of the KS theorem that rules out hidden variable theories with outcome assignments in the set {0, p, 1 − p, 1} for p ∈ [0, 1/d) ∪ (1/d, 1/2]. The case p = 1/2 is especially physically significant. We show that in this case the result rules out (consistent) hidden variable theories that are fundamentally binary, i.e., theories where each measurement has fundamentally at most two outcomes (in contrast to the single deterministic outcome per measurement ruled out by KS). We present a device-independent application of this generalised KS theorem by constructing a two-player non-local game for which a perfect quantum winning strategy exists (a Pseudo-telepathy game) while no perfect classical strategy exists even if the players are provided with additional no-signaling resources of PR-box type (with marginals in {0, 1/2, 1}).
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.
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.
We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not only can we quantify the idea of finite-precision measurement within our theory, but we can also suggest a possible resolution of the Meyer-Mermin debate on the impact of finite-precision measurement on the Kochen-Specker theorem.
Quantum processes cannot be reduced, in a nontrivial way, to classical processes without specifying the context in the description of a measurement procedure. This requirement is implied by the Kochen--Specker theorem in the outcome-deterministic case and, more generally, by the Gleason theorem. The latter establishes that there is only one noncontextual classical model compatible with quantum theory, the one that trivially identifies the quantum state with the classical state. However, this model requires a breaking of the unitary evolution to account for macroscopic realism. Thus, a causal classical model compatible with the unitary evolution of the quantum state is necessarily contextual to some extent. Inspired by well-known results in quantum communication complexity, we consider a particular class of hidden variable theories by assuming that the amount of information about the measurement context is finite. Aiming at establishing some general features of these theories, we first present a generalized version of the Gleason theorem and provide a simple proof of it. Assuming that Gleason's hypotheses hold only locally for small changes of the measurement procedure, we obtain almost the same conclusion of the original theorem about the functional form of the probability measure. An additional constant and a relaxed property of the density operator are the only two differences from the original result. By this generalization of the Gleason theorem and the assumption of finite information for the context, we prove that the probabilities over three or more outcomes of a projective measurement must be linear functions of the projectors associated with the outcomes, given the information on the context.