Alice and Bob share and play the CHSH game with it

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Nonlocality: EPR, Bell inequalities and the CHSH game

Problem

Alice and Bob share ∣Φ+⟩=(∣00⟩+∣11⟩)/2|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2 and play the CHSH game with it. Each measures along a direction in the xx-zz plane, so for settings at Bloch angles θa\theta_a and θb\theta_b (measured from the zz-axis) the correlation is E(θa,θb)=+cos⁡(θa−θb)E(\theta_a,\theta_b)=+\cos(\theta_a-\theta_b). On input x=0x=0 Alice measures at a0=0∘a_0=0^\circ, on x=1x=1 at a1=90∘a_1=90^\circ; on input y=0y=0 Bob measures at b0=30∘b_0=30^\circ, on y=1y=1 at b1=−60∘b_1=-60^\circ. Each answers the bit 00 for outcome +1+1 and 11 for outcome −1-1. ((a)) Compute the CHSH combination S=E(a0,b0)+E(a0,b1)+E(a1,b0)−E(a1,b1)S=E(a_0,b_0)+E(a_0,b_1)+E(a_1,b_0)-E(a_1,b_1). ((b)) Using pwin=12+S/8p_{\rm win}=\tfrac12+S/8, find the probability that this strategy wins, and check that it lies between the classical value 3/43/4 and the quantum maximum 0.8540.854.

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