Contradiction of Quantum Mechanics with Local Hidden Variables for Continuous Variable Quadrature Phase Amplitude Measurements

We demonstrate a contradiction of quantum mechanics with local hidden variable theories for continuous variable quadrature phase amplitude (“position” and “momentum”) measurements, by way of a violation of a Bell inequality. For any quantum state, this contradiction is lost for situations where the quadrature phase amplitude results are always macroscopically distinct. We show that for optical realisations of this experiment, where one uses homodyne detection techniques to perform the quadrature phase amplitude measurement, one has an amplification prior to detection, so that macroscopic fields are incident on photodiode detectors. The high efficiencies of such detectors may open a way for a loophole-free test of local hidden variable theories. In 1935 Einstein, Podolsky and Rosen [1] presented an argument for the incompleteness of quantum mechanics. The argument was based on the validity of two premises: no actionat-a-distance (locality) and realism. Bell [2] later showed that the predictions of quantum mechanics are incompatible with the premises of local realism (or local hidden variable theories). Experiments [3] based on Bell’s result support quantum mechanics, indicating the failure of local hidden variable theories. One feature appears characteristic of all the contradictions of quantum mechanics with local hidden variables studied to date. The measurements considered have discrete outcomes, for example being measurements of spin or photon number. By this we mean specifically that the eigenvalues of the appropriate system hermitian operator, which represents the measurement in quantum mechanics, are discrete. In this paper we show how the predictions of quantum mechanics are in disagreement with those of local hidden variable theories for a situation involving continuous quadrature phase amplitude (“position” and “momentum”) measurements. By this we mean that the quantum predictions for the probability of obtaining results x and p for position and momentum (and