Experimental Demonstration of Fermion Spin Correlations

Bell's Theorem places limits on correlations between local measurements of particles whose properties are independent of measurement. In particular, Bell's Theorem limits the mean product of binary spin measurements at $45^\circ$ (or $135^\circ$) separation to be $|P| \le 0.5$. However, Bell's Theorem is not valid for spins sampled on a spherical distribution because the density of sampled states depends on the sampling location. We model spin-1/2 fermions as azimuthally symmetric spherical standing waves with one hemisphere of spin up and one hemisphere of spin down. We experimentally determine the spin correlation for $45^\circ$ separation by randomly placing two points with fixed separation on a ball marked with lines of latitude. The normalized product of spins is $+1$ if the two points are on the same side of the equator, and $-1$ if the points are on opposite sides of the equator. The expected correlation (mean product) is $P=1-45/90 = 0.5$. Correcting for the lack of azimuthal symmetry in the experimental ball increases the expected correlation in our model to approximately $P = \cos 45^\circ \approx 0.71$, inconsistent with Bell's Theorem but consistent with experimental measurements on entangled spin-1/2 fermions.