Violation of Bell’s inequality with continuous spatial variables

The Einstein-Podolsky-Rosen (EPR) argument revealed the paradoxical properties of a two-particle system entangled continuously in the spatial parameter. Yet a direct test of quantum nonlocality exhibited by this state, via a violation of Bell’s inequality, has not been forthcoming. In this paper, we identify and construct experimental arrangements comprising simple optical components, without nonlinearities or moving parts, that implement operators in the spatial-parity space of single-photon fields that correspond to the familiar Pauli spin operators. We achieve this by first establishing an isomorphism between the single-mode multiphoton electromagnetic-field space spanned by a Fock-state basis and the single-photon multimode electromagnetic-field space spanned by a spatial-eigenmode basis. We then proceed to construct a Hilbert space with a two-dimensional basis of spatial even-odd parity modes. In particular, we describe an arrangement that implements a rotation in the parity space of each photon of an entangled-photon pair, allowing for a straightforward experimental test of Bell’s inequality using the EPR state. Finally, the violation of a Bell inequality is quantified in terms of the physical parameters of the two-photon source.