Exclusivity principle and the quantum bound of the Bell inequality

We show that, for general probabilistic theories admitting sharp measurements, the exclusivity principle together with two assumptions exactly singles out the Tsirelson bound of the Clauser-Horne-Shimony-Holt Bell inequality.

Figure 1

A key point of the proof is the observation that extra measurements may be carried out after sharp measurements used for a CHSH Bell experiment. Specifically, after Alice has measured $A_i$ and ${\mathrm{Alice}}^{\prime }$ has measured $A_i^{\prime }$, Alice and ${\mathrm{Alice}}^{\prime }$ can also measure $A_i{A}_i^{\prime }$.