Simplified quantum optical Stokes observables and Bell's theorem

We discuss a simplified form of Stokes operators for quantum optical fields that involve the known concept of binning. Behind polarization analyzer photon numbers (more generally intensities) are measured. We have two outputs, say, for horizontal and vertical polarization. If the value obtained in horizontal output is greater than in vertical one we put 1. Otherwise, we put - 1. For equal photon numbers, we put 0. Such observables do not have all properties of the Stokes operators, but can be employed in Bell type measurements, involving polarization analyzers. They are especially handy for states of undefined number of photons, e.g. squeezed vacuum and their realisation is intuitive. We show that our observables can lead to quite robust violations of associated Bell inequalities. We formulate a strongly supported numerically conjecture that one can observe with this approach violations of local realism for the four mode squeezed vacuum for all pumping powers (i.e. gain values).