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arXiv:0705.1385 · Physical Review Letters

Bell Inequalities for Continuous-Variable Correlations

Eric G. Cavalcanti, C. J. Foster, M. D. Reid, P. D. Drummond

Published Nov 21, 2007• 108 citations
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Abstract

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous-variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.

DOI
10.1103/physrevlett.99.210405

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We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous-variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.
In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.
A common experimental strategy for demonstrating nonclassical correlations is to show violation of a Bell inequality by measuring a continuously emitted stream of entangled photon pairs. The measurements involve the detection of photons by two spatially separated parties. The detection times are recorded and compared to quantify the violation. The violation critically depends on determining which detections are coincident. Because the recorded detection times have ``jitter,'' coincidences cannot be inferred perfectly. In the presence of settings-dependent timing errors, this can allow a local-realistic system to show apparent violation, the so-called ``coincidence loophole.'' Here, we introduce a family of Bell inequalities based on signed, directed distances between the parties' sequences of recorded time tags. Given that the time tags are recorded for synchronized, fixed observation periods and that the settings choices are random and independent of the source, violation of these inequalities unambiguously shows nonclassical correlations violating local realism. Distance-based Bell inequalities are generally useful for two-party configurations where the effective size of the measurement outcome space is large or infinite. We show how to systematically modify the underlying Bell functions to improve the signal-to-noise ratio and to quantify the significance of the violation.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.