Every measurement determines a single value as its outcome, and yet quantum mechanics predicts it only probabilistically. The Kochen-Specker theorem and Bell's inequality are often considered to reject a realist view but favor a skeptical view that measuring an observable does not mean ascertaining the value that it has, but producing the outcome, having only a personal meaning. However, precise analysis supporting this view is unknown. Here, we show that a quantum mechanical analysis turns down this view. Supposing that two observers simultaneously measure the same observable, we can well pose the question as to whether they always obtain the same outcome, or whether the probability distributions are the same, but the outcomes are uncorrelated. Contrary to the widespread view in favor of the second, we shall show that quantum mechanics predicts that only the first case occurs. We further show that any measurement establishes a time-like entanglement between the observable to be measured and the meter after the measurement, which causes the space-like entanglement between the meters of different observers. We also show that our conclusion cannot be extended to measurements of so-called 'generalized' or 'unsharp' observables, suggesting a demand for reconsidering the notion of observables in foundations of quantum mechanics.
This paper argues that physical quantities should be reconstructed not as non-contextual intrinsic attributes with measurement-independent determinate values, but as equivalence classes of measurement procedures. Bell–CHSH violations first motivate the problem: they show that Bell-type local causality/factorizability, measurement independence, and the relevant form of outcome determinism or value-definiteness cannot be jointly maintained, without by themselves refuting realism as such. Kochen–Specker contextuality supplies the stronger premise for the sharp-observable case: no global, non-contextual, dispersion-free value assignment to projection-valued observables can preserve the functional relations among compatible observables in Hilbert spaces of dimension at least three. The positive proposal uses algebraic quantum theory to identify physical quantities with classes of measuring procedures that yield the same outcome statistics over a full or tomographically complete state domain. This reconstruction preserves the classical deterministic limit while rejecting the stronger claim that measured quantities are context-independent values possessed independently of measurement. The analysis is restricted to sharp observables and projection-valued measurements; POVMs and unsharp measurements are noted as extensions. The resulting view is realist about an observer-independent world but non-intrinsic about empirical quantities, and it situates Wheeler's "It from Bit" as an interpretive corollary rather than as a foundational premise.
A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here, compatible measurements are those that can be performed simultaneously or in any order without disturbance. This conflict is generically called quantum contextuality. In this article, we present an introduction to this subject and its current status. We review several proofs of the Kochen-Specker theorem and different notions of contextuality. We explain how to experimentally test some of these notions and discuss connections between contextuality and nonlocality or graph theory. Finally, we review some applications of contextuality in quantum information processing.
In this paper, we examine measurement using the Q-based, objective-field model for quantum mechanics. Schrodinger considered a microscopic system prepared in a superposition of states which is then coupled to a macroscopic meter. We analyze the entangled meter and system, and measurements on it, by solving forward-backward stochastic differential equations for real amplitudes
x(t) and
p(t) that correspond to the phase-space variables of the Q function of the system at a time
t. We model the system and meter as single-mode fields, and measurement of
x^ by amplification of the amplitude
x(t). Our conclusion is that the outcome for the measurement is determined at (or by) the time
tm, when the coupling to the meter is complete, the meter states being macroscopically distinguishable. There is consistency with macroscopic realism. By evaluating the distribution of the amplitudes
x and
p postselected on a given outcome of the meter, we show how the
Q-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes
x and
p are too narrow to comply with the uncertainty principle, ruling out that the distribution represents a quantum state. We conclude that the collapse of the wavefunction occurs as a two-stage process: First there is an amplification that creates branches of amplitudes
x(t) of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by
x(t) once amplified, explaining Born's rule. Second, the distribution that determines the final collapse is the state inferred for the system conditioned on the outcome of the meter: information is lost about the meter, in particular, about the complementary variable
p.
This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as “non-classical”. The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell’s criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics.
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. As a result, the model satisfies the Bell inequality; there are no so-called violations. We extend this model to include a hidden variable. This extended model is not Bell-separable. This non-separability implies that the model is non-deterministic or non-local (or both).
The circumstances of measurement have more direct significance in quantum than in classical physics, where they can be neglected for well-performed measurements. In quantum mechanics, the dispositions of the measuring apparatus-plus-environment of the system measured for a property are a non-trivial part of its formalization as the quantum observable. A straightforward formalization of context, via equivalence classes of measurements corresponding to sets of sharp target observables, was recently given for sharp quantum observables. Here, we show that quantum contextuality, the dependence of measurement outcomes on circumstances external to the measured quantum system, can be manifested not only as the strict exclusivity of different measurements of sharp observables or valuations but via quantitative differences in the property statistics across simultaneous measurements of generalized quantum observables, by formalizing quantum context via coexistent generalized observables rather than only its subset of compatible sharp observables. Here, the question of whether such quantum contextuality follows from basic quantum principles is then addressed, and it is shown that the Principle of Indeterminacy is sufficient for at least one form of non-trivial contextuality. Contextuality is thus seen to be a natural feature of quantum mechanics rather than something arising only from the consideration of impossible measurements, abstract philosophical issues, hidden-variables theories, or other alternative, classical models of quantum behavior.