Alice and Bob share

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Information & Technology · Quantum states, measurement and entanglement

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

Problem

Alice and Bob share ∣ψ(α)⟩=cos⁡α∣00⟩+sin⁡α∣11⟩|\psi(\alpha)\rangle=\cos\alpha|00\rangle+\sin\alpha|11\rangle. Alice measures ZZ or XX; Bob measures B(ϕ)=cos⁡ϕ Z+sin⁡ϕ XB(\phi)=\cos\phi\,Z+\sin\phi\,X or B(−ϕ)=cos⁡ϕ Z−sin⁡ϕ XB(-\phi)=\cos\phi\,Z-\sin\phi\,X, for some angle ϕ\phi still to be chosen. (a) Show that ⟨ZZ⟩=1\langle ZZ\rangle=1, ⟨XX⟩=sin⁡2α\langle XX\rangle=\sin2\alpha and ⟨ZX⟩=⟨XZ⟩=0\langle ZX\rangle=\langle XZ\rangle=0 on ∣ψ(α)⟩|\psi(\alpha)\rangle. (b) Hence show that the CHSH combination, with Alice's settings Z,XZ,X and Bob's settings B(ϕ),B(−ϕ)B(\phi),B(-\phi), is S(ϕ)=2cos⁡ϕ+2sin⁡2α sin⁡ϕS(\phi)=2\cos\phi+2\sin2\alpha\,\sin\phi. (c) Maximise S(ϕ)S(\phi) over ϕ\phi and confirm the quoted result Smax=21+sin⁡22αS_{\rm max}=2\sqrt{1+\sin^22\alpha}. (d) At α=π/8\alpha=\pi/8, give the optimal ϕ\phi in degrees and the corresponding SmaxS_{\rm max}.

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