Bloch sphere colourings and Bell inequalities

We consider here the predictions of quantum theory and local hidden variables for the correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by a given angle \theta. The predictions of local hidden variable models for projective measurements on qubits correspond to binary colourings of the Bloch sphere with antipodal points oppositely coloured. We prove Bell inequalities separating the predictions of all local hidden variable models from the singlet correlations predicted by quantum theory for all \theta in the range 0 < \theta < \pi/3. We raise and explore the possibility of proving stronger Bell inequalities directly from optimization results on sphere colourings. In particular, we explore strong and weak forms of the hemispherical colouring maximality hypothesis (HCMH) that, for a continuous range of \theta > 0, the maximum LHV anti-correlation is obtained by assigning to each qubit a colouring with one hemisphere black and the other white. Our results show that hemispherical colourings are near-optimal for small \theta; we also describe numerical tests consistent with the HCMH that bound the range of \theta. Finally, we note proofs of related results for binary colourings of R^n.