A general algebraic formalism for studying contextual hidden variable theories is developed. The contextuality is understood as a manifestation of an inadequacy in the algebra of quantum observables for the complete description of the system. It is shown that it is always possible to 'improve' the algebra of quantum observables, explicitly paying attention to contexts, such that this improved algebra becomes a base for a hidden variable theory.
A general algebraic formalism for studying contextual hidden variable theories is developed. The contextuality is understood as a manifestation of an inadequacy in the algebra of quantum observables for the complete description of the system. It is shown that it is always possible to 'improve' the algebra of quantum observables, explicitly paying attention to contexts, such that this improved algebra becomes a base for a hidden variable theory.
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.
Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability conditions implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed.
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.
Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability condition implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed. INTRODUCTION In most of the discussions of Bell's Inequalities little notice is taken of the historical circumstances of their discovery. J. S. Bell was told by J. Jauch of the work of A. Gleason [1957] which showed that simultaneous values cannot be assigned to all observables of a quantum mechanical system in a way that respects their algebraic structure (except in the simple case in which the relevant Hilbert space has dimension less than three). Gleason's theorem thus precludes a type of hidden variables theory that has come to be called ‘noncontextual’. The question of agreement between the statistical predictions of quantum mechanics and those of a noncontextual hidden variables theory does not even arise, since such a theory cannot be consistently formulated.
We present a simple way of incorporating the structure of contextual extensions into quantum gravity models. The contextual extensions of C∗-algebras, originally proposed for contextual hidden variables, are generalized to the cones indexed by the contexts and their limit in a category. By abstracting the quantum gravity models as functors, we study the contextual extensions as the categorical limits of these functors in several quantum gravity models. Such contextual extensions of quantum gravity models are useful for building topos-theoretic models of quantum gravity.
While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.