Phase averaging of a polarization Bell state ($\mathcal{B}=\sqrt{2}(1+\mathrm{sinc}{\phi }_0)$)

Uniformly averaging $(|01⟩+e^{i\phi }|10⟩)/\sqrt{2}$ over $|\phi |\le {\phi }_0$ leaves the two populations unchanged and multiplies the off-diagonal coherences by $\mathrm{sinc}{\phi }_0=\sin {\phi }_0/{\phi }_0$. For real-plane analyser settings giving the pure-state Bell signal $\sqrt{2}(1+\cos \phi )$, linearity of expectation gives the displayed mixed-state signal. Its local-bound violation requires $\mathrm{sinc}{\phi }_0>\sqrt{2}-1$, or $0\le {\phi }_0<2.092751\dots$ radians. This threshold concerns fixed measurement settings rather than all possible tests of entanglement.