Bell inequalities with continuous angular variables

Abstract

We consider bipartite quantum systems characterized by a continuous angular variable θ ∈ [−π, π[, representing, for instance, the position of a particle on a circle. We show how to reveal non-locality on this type of system using inequalities similar to 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 non locality on a variety of quantum systems described by continuous variables.

Continuous discretization of infinite-dimensional Hilbert spaces

“…φi , analogously to CHSH Bell-type inequalities [5,11]. The present results generalize those derived for continuous variables operators with a bounded spectrum [26][27][28] to arbitrary continuous or discrete variables systems, irrespectively of their dimension or spectral properties. Belltype inequalities involving correlations between SU (d) operators [29,30] can also be generalized through the present formalism by using correlations between Γ(d) α observables.…”

“…(29) can be written asσ 3 (θ,k), i.e., aθ,k dependent Pauli matrix. Analogously, the other 2 {θ,k} dependent Pauli matrices can be defined, up to a global phase in the basis state (see (28)), as in Eq. (3) of the main text:…”

Experimental violation of a Bell-like inequality with optical vortex beams

“…However, this inequality, in different forms, has been used in recent times for other kind of variables, as long as one deals with two-valued positive operator-valued measurements [33,34]. The resulting inequality has already been applied to continuous angular variables [35], as is the case of a particle moving on a ring.…”

Violation of an extended Wigner inequality with high-angular-momentum states

“…It is worthwhile to mention in passing a recent paper that might be helpful in this situation; based on CSHS-type inequalities for continuous periodic variables: Ref. [20].…”