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.
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.
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.
Abstract In chapter 12 we give an overview of the historical development of hidden-variable theories aiming to attribute ‘hidden’ definite outcomes to any potential measurement on a quantum system. We start with a discussion of von Neumann’s assumption on hidden-variable models, which was later realized not to hold by Hermann, Kochen, Specker, and Bell, going into detail on Bell’s refutation of von Neumann’s arguments. We then turn to the notion of contextuality, which severely restricts hidden-variable models: we discuss Gleason’s theorem before stating and proving the famous Kochen-Specker theorem, as well as the simpler versions of Peres’ nonet and Mermin’s pentagram. These results show that quantum mechanics is not compatible with the assignment of definite measurement outcomes to all possible measurements, and such assignments can hence only be made once a restricted set of observables, a context, is selected
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.
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.
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}).