Derivation and experimental violation of Bell-like inequalities involve the measurement of incompatible observables. Simple complementarity forbids the existence of such joint probability distribution. Moreover, the measurement of incompatible observables requires different experimental procedures, which no necessarily must share a common joint statistics. In this work, we avoid these difficulties by proposing a joint simultaneous measurement experiment. We can obtain the exact individual statistics of all the observables involved in the Bell inequalities after a suitable data inversion. A lack of positivity or any other pathology of the so retrieved joint distribution is then a signature of nonclassical behavior. PACS numbers: 42.25.-p Wave optics 42.50.Dv Quantum state engineering and measurements 03.65.Ud Entanglement and quantum nonlocality 003.65.Ta Foundations of quantum mechanics; measurement theory, 42.50.Xa Optical tests of quantum theory
We use the uncertainty relation between the operators associated with the total number of particles and with the relative phase of two bosonic modes to construct entanglement and Einstein-Podolsky-Rosen steering criteria. These can be tested experimentally in a variety of systems, such as optical fields, Bose-Einstein condensates, and mechanical oscillators. While known entanglement criteria involving the phase observable typically require us to perform interference measurements by recombining the two systems, our criteria can be tested through local measurements at two spatially distinct positions to investigate the nonlocal nature of quantum correlations. We present simple examples where our criteria are violated and show their robustness to noise. Apart from being useful for state characterization, they might find application in quantum information protocols, for example, based on number-phase teleportation.
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