We consider bipartite quantum systems characterized by a continuous angular variable $\theta$, representing, for instance, the position of a particle on a circle or in a periodic lattice. For this type of system, we show how to violate inequalities similar to Clauser-Horne-Shimony-Holt (CHSH) ones, originally derived for bipartite spin-$1/2$-like systems. Such inequalities involve correlated measurement of continuous angular functions and are equivalent to the continuous superposition of CHSH inequalities acting on bidimensional subspaces of the infinite dimensional Hilbert space. As an example, we discuss in detail one application of our results, and we derive inequalities based on orientation correlation measurements. The introduced Bell-type inequalities open the perspective of new and simpler experiments to test the assumptions of locality, realism, or freedom of choice in a variety of quantum systems described by continuous variables. We discuss in detail a possible implementation using the orbital angular momentum of photons.

• Received 15 September 2011

DOI:https://doi.org/10.1103/PhysRevA.86.052107