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Foundations of Physics

Joint probabilities of noncommuting operators and incompleteness of quantum mechanics

A. O. Barut, Mirjana Božić, Z. Marić

Published Oct 1, 1988• 9 citations
Publisher
DOI
10.1007/bf01909935

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Noncommuting observables cannot be simultaneously measured; however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables. We find that the Quasi-Bell inequalities have a quantum to classical violation as high as 32 on two qubit, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics and provides an insightful generalization of Bell inequalities.
It is well known that noncommuting observables cannot be simultaneously measured in quantum theory. However, under hidden variable models, such observables must simultaneously hold premeasurement values. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fr\'echet bounds, finding that the joint probabilities may be negative. We then use the symmetrization heuristic to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct generalized Bell inequalities with a quantum to classical violation of \frac{3}{2}. The result demonstrates the importance of noncommutativity in the nonclassicality of quantum mechanics. However, the additional symmetrization assumption makes its conclusions weaker than traditional Bell inequalities.
The collapse of a quantum state is a mathematical sleight of hand that allows the construction of a joint probability density even for operators that are noncommutative. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. A collapse picture, in which a quantum state collapses after each measurement, can therefore be equivalent to a no-collapse picture that uses a different state and different, mutually commutative, Quantum Non-Demolition operators to construct the same joint probability density for consecutive measurements. The collapse of a state is a mathematical construction that can also be a useful tool when using the Koopman formalism for classical mechanics and signal analysis.