Bloch-sphere colorings and Bell inequalities

Abstract

We consider the quantum and local hidden variable (LHV) correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by an angle $\ensuremath{\theta}$. Local hidden variables predict binary colorings of the Bloch sphere with antipodal points oppositely colored. We prove Bell inequalities separating the LHV predictions from the singlet quantum correlations for $\ensuremath{\theta}\ensuremath{\in}(0,\frac{\ensuremath{\pi}}{3})$. We raise and explore the hypothesis that, for a continuous range of $\ensuremath{\theta}>0$, the maximum LHV anticorrelation is obtained by assigning to each qubit a coloring with one hemisphere black and the other white.