Testing Hidden Variable Theory with CHSH-Bell Inequality Experiment

Abstract

Introduction and Theory

In their influential paper, Einstein, Podolsky, and Rosen (EPR) explored an ostensible paradox that arises from quantum mechanics [1]. They concluded that quantum mechanics could not be a complete theory, as it does not allow two non commuting operators and their physical quantities to have simultaneous realities. Additionally, they rejected the possibility of entanglement as described by quantum mechanics as being a realistic physical phenomena, and that though the physical prediction was correct, it could not be explained solely with quantum mechanical theories.

However, as Bell derived and others have demonstrated, it is possible to measure the correlation between two particles against Bell’s inequality, thus showing the incompatibility of local hidden variable theory with physical reality. In our experiment, however, we followed Clauser, Horne, Shimony, and Holt (CHSH) [2], in which they derived a more practical way to measure the inequality by using photon polarization instead of electron spin.

Using this photon polarization basis, the value for the inequality is given by [3]

\[2\geq \left| S \right| = \left| E(a,b)-E(a,b^\prime)+E(a^\prime,b)+E(a^\prime,b^\prime) \right|\]

where \((a,b,a^\prime,b^\prime)\) are polarization angles. The correlation between two angles is given by [3]

\[E(a,b)=\frac{N(a,b)+N(a_\perp,b_\perp)-N(a_\perp,b)-N(a,b_\perp)}{N(a,b)+N(a_\perp,b_\perp)+N(a_\perp,b)+N(a,b_\perp)}\]

where the perpendicular angles are rotated by \(\frac{\pi}{2}\). Thus, only 16 separate measurements had to be taken for the \(\psi_+\) and \(\psi_-\) states. The angles were chosen such that the value of \(S\) was maximized to increase the likelihood of violating the CHSH-Bell Inequality. For \(\psi_+\) and \(\psi_-\), the values of \((a,b,a^\prime,b^\prime) = (0,67.5,45,22.5)\), and \((-45,-22.5,0,22.5)\), respectively.

[fig:Figure 1] Experimental Setup [4]