Fig. 1: CHSH test setup.

A central source distributes quantum states ρ_AB to Alice and Bob, who interact with their measurement devices by providing binary inputs (x and y) and recording binary outputs (a and b) to estimate their CHSH winning probability, ω. As an example, the Tsirelson’s bound \(\omega =(2+\sqrt{2})/4\) is reached if \({\rho }_{{{{\rm{AB}}}}}=\left\vert {\Phi }^{+}\right\rangle \left\langle {\Phi }^{+}\right\vert {{\Phi }^{+}}_{{{{\rm{AB}}}}}\) (with \({\left\vert {\Phi }^{+}\right\rangle }_{{{{\rm{AB}}}}}=({\left\vert 00\right\rangle }_{{{{\rm{AB}}}}}+{\left\vert 11\right\rangle }_{{{{\rm{AB}}}}})/\sqrt{2}\)) and the inputs x and y determine the measurement of the observables A_x and B_y, where A₀ = σ_Z, A₁ = σ_X, \({B}_{0}=({\sigma }_{{{{\rm{Z}}}}}+{\sigma }_{{{{\rm{X}}}}})/\sqrt{2}\) and \({B}_{1}=({\sigma }_{{{{\rm{Z}}}}}-{\sigma }_{{{{\rm{X}}}}})/\sqrt{2}\) (σ_Z and σ_X being Pauli matrices). In this ideal example, Alice records output a upon observation of the eigenvalue (−1)^a, and similarly for Bob.