Experimental violation of Tsirelson's bound by Maxwell fields

In analogy with quantum optics it is possible to impose nonseparability between different degrees of freedom of an optical beam. The resulting correlations between these degrees of freedom can be investigated with correlations functions traditionally employed in quantum mechanics, such as the well-known Clauser-Horne-Shimony-Holt (CHSH) correlation function. In this paper we present results achieving a maximal violation of Tsirelson's bound on CHSH correlations between spatial and polarization degrees of freedom of classical (Maxwell) fields. We describe the theoretical method, based on the realization of a nonunitary gate, and then proceed to its experimental implementation carried out with classical optical techniques. Our approach relies on the realization at the level of classical Maxwell fields of a so-called POVM (positive operator valued measure) which is traditionally discussed in the realm of quantum physics.

Figure 1

Parity analyzer and its dual counterpart based on Mach-Zehnder interferometry.

Figure 2

(a) Realization of Eq. (49) with a nonunitary gate and (b) parity analyzer [Eq. (55)] in the double beam configuration.

Figure 3

Optical setup for preparing a Bell state which can be modified by a tunable device denoted bifunctional box (BB) and analyzed (i) in polarization with a polarizing beam splitter cube (PBC) and (ii) in parity (PA*) with a parity analyzer PA2 or a nonunitary gate PA1 (see Fig. 4). $P$ is a linear polarizer. SF/BE is a spatial filter used to spatially clean and expand the laser beam. $2H$ is an aluminium plate perforated by two holes symmetric relative to the optical axis. $\frac{\lambda }{2}$ is a half wave plate. The inset “Polarization States” shows the beams ($\left|\text{up}\right.〉$ and $\left|\text{down}\right.〉$) at each stage.

Figure 4

Nonunitary gate (PA1) and parity analyzer (PA2).

Figure 5

(a)–(d) Experiments realized with an empty bifunctional box (BB). (e)–(h) Experiments realized with the bifunctional box (BB) being a half wave plate in order to rotate polarizations by an angle of $\frac{\pi }{2}$. (a),(e) and (b),(f) are parity measurements (PA2 in Fig. 4) for respectively the vertical and horizontal polarizations (3 and 4 in Fig. 3). (c),(g) and (d),(h) are nonunitary measurements (PA1 in Fig. 4) for respectively the vertical and horizontal polarizations (3 and 4 in Fig. 3).