Answers to First Midterm Examination 1. Two spin 1/2 particles ...

having a known probability of occurrence, i.e., an ensemble. Describe Alice’sensemble and compute the density matrix ρ of this ensemble.Alice’s ensemble is |ϕ + 〉 = (2 |↑〉 + i |↓〉)/ √ 5 with probability p + = 5/6 and|ϕ − 〉 = |↓〉 with probability p − = 1/6. The density matrix for this ensemble isρ = p + |ϕ + 〉 〈ϕ + | + p − |ϕ − 〉 〈ϕ − |= 5 16 5 (2 |↑〉 + i |↓〉)(2 〈↑| − i 〈↓|) + 1 |↓〉 〈↓|6= 1 ()4 |↑〉 〈↑| − 2i |↑〉 〈↓| + 2i |↓〉 〈↑| + 2 |↓〉 〈↓|6Or in matrix formρ = 1 3[ 2 −ii 1].(d) Is ρ = ρ A (where ρ is the density matrix computed in part (c), and ρ A is thereduced density matrix in part (a))? What does this equality or inequality mean?Yes, ρ = ρ A . Any actions taken by Bob cannot change the density matrix seenby Alice. This implies that no communication can be done by entangled pairs.2. (a) Let ˆn be a unit vector on the xy-plane in the form ˆn = cos φˆx + sin φŷ where φ isone of the spherical angles of ˆn (the other is θ = π/2). Find σ n = ⃗σ · ˆn.[ ] 0 e−iφσ n = cos φ σ x + sin φ σ y =e iφ .0(b) Find σ n |↑〉 and σ n |↓〉. (You should use the matrix form of σ n above to computethese. But the ket expressions in part (b) will be more useful below.)σ n |↑〉 = e iφ |↓〉 , σ n |↓〉 = e −iφ |↑〉 .Consider the following GHZ state of three spins possessed by Alice, Bob and Charlie|GHZ〉 = 1 √2(|↑↑↑〉 − |↓↓↓〉)= 1 √2(|↑〉 A⊗ |↑〉 B⊗ |↑〉 C− |↓〉 A⊗ |↓〉 B⊗ |↓〉 C) .(Note the coefficient of the second term which is different from the one we used inclass.) Let ˆn A , ˆn B and ˆn C be three unit vectors on the xy-plane having respectiveangles φ A , φ B and φ C .(c) Find (σ AnA ⊗ σ BnB ⊗ σ CnC ) |GHZ〉.(σ AnA ⊗ σ BnB ⊗ σ CnC ) |GHZ〉 = 1 √2(ei(φ A +φ B +φ C ) |↓↓↓〉 − e −i(φ A+φ B +φ C ) |↑↑↑〉 ) .2