Alice holds qubit 1 in the state with Bloch vector

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Problem

Alice holds qubit 1 in the state ∣ψ⟩|\psi\rangle with Bloch vector (0.6,0,0.8)(0.6, 0, 0.8), and shares the Bell pair ∣Φ+⟩23|\Phi^+\rangle_{23} with Bob, who holds qubit 3. Alice performs the standard teleportation measurement on qubits 1 and 2 in the Bell basis ∣βjk⟩=(ZjXk⊗I)∣Φ+⟩|\beta_{jk}\rangle = (Z^jX^k\otimes I)|\Phi^+\rangle. ((a)) Before Alice's two classical bits reach Bob, what is the Bloch vector of qubit 3, averaged over the four equally likely outcomes? State which no-signalling fact from Chapter 3 this reproduces. ((b)) Alice's actual outcome is (j,k)=(1,1)(j,k) = (1,1). Find the Bloch vector of qubit 3 at that moment, before Bob applies any correction. ((c)) Bob now applies the correction ZjXkZ^jX^k appropriate to this outcome. Find the resulting Bloch vector of qubit 3 and check it against ∣ψ⟩|\psi\rangle. ((d)) After Alice's measurement, qubits 1 and 2 are left in the Bell state ∣β11⟩|\beta_{11}\rangle. Identify this state in the ∣Φ±⟩,∣Ψ±⟩|\Phi^\pm\rangle, |\Psi^\pm\rangle notation, compute the eigenvalues of its partial transpose, and compute its CHSH value using the Werner-state formula 22 w2\sqrt2\,w at the appropriate ww. State whether qubits 1 and 2 are entangled and whether they violate CHSH.

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