Bell inequalities for general quantum observables

Wiesław Laskowski, Tomasz Paterek, Marek Żukowski +1 more

We derive tight Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give a necessary and sufficient condition for a general quantum state to violate the new inequalities. The inequalities are violated by some classes of states, for which all standard Bell's inequalities with two measurement settings per observer are satisfied.

H. E. Andås

Liberato De, Augusto Garuccio

A general approach to Bell-type inequalities for trichotomic observables is presented. With this formalism, which explicitly includes no-counts events, we show that the limits of all Bell-type inequalities are functions of the supplementary assumptions. This approach enables us to include in the causal and local theory the same assumptions used in the quantum-mechanical analysis of an experiment, and to deduce the inequality that must be satisfied by any local realistic model incorporating these assumptions. We prove that, using the quantum-mechanical assumption of ``random nondetection'' in the causal and local approach, a generalized Bell-type inequality is deduced whose upper limit can be violated by the quantum-mechanical predictions for symmetrical experimental setups involving detectors with quantum efficiency greater than 0.811. \textcopyright{} 1996 The American Physical Society.

Omar E. Gamel, Graham R. Fleming

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.

B. S. Cirel'son