Abstract
Bell inequality testing, a well-established method to demonstrate quantum non-locality between remote two-partite entangled systems, is playing an important role in the field of quantum information. The extension to high-dimensional entangled systems, using the so-called Bell-CGLMP inequality, points the way in measuring joint probabilities, the kernel block to construct high dimensional Bell inequalities. Here we show that in theory the joint probability of a two-partite system entangled in a Hilbert space can be measured by choosing a set of basis vectors in its dual space that are related by a Fourier transformation. We next propose an experimental scheme to generate a high-dimensional position-entangled two-photon state aided by a combination of a multiple-slit and a 4 f system, and describe a method to test Bell's inequality using correlated diffraction. Finally, we discuss in detail consequences of such Bell-test violations and experimental requirements.