= Phase averaging of a polarization Bell state
{title2=$\mathcal B=\sqrt2(1+\operatorname{sinc}\varphi_0)$}
Uniformly averaging $(|01\rangle+e^{i\varphi}|10\rangle)/\sqrt2$ over $|\varphi|\leq\varphi_0$ leaves the two populations unchanged and multiplies the off-diagonal coherences by $\operatorname{sinc}\varphi_0=\sin\varphi_0/\varphi_0$. For real-plane analyser settings giving the pure-state Bell signal $\sqrt2(1+\cos\varphi)$, linearity of expectation gives the displayed mixed-state signal. Its local-bound violation requires $\operatorname{sinc}\varphi_0>\sqrt2-1$, or $0\leq\varphi_0<2.092751\ldots$ radians. This threshold concerns fixed measurement settings rather than all possible tests of entanglement.
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