Let A and B be non-identical particles. A 's Hilbert space is spanned by { | 0 ⟩ , | 1 ⟩ } , the eigenstates of an observable H A . B 's Hilbert space is spanned by { | p ⟩ , | q ⟩ , | r ⟩ } , the eigenstates of an observable H B . The normalized two-particle state is | ψ ⟩ = 1 2 | 0 ⟩ | p ⟩ + 1 6 | 1 ⟩ | q ⟩ + C | 0 ⟩ | q ⟩ + 1 3 | 1 ⟩ | r ⟩ . (a) Determine C . (1 marks) (b) Suppose we perform a H A measurement, and obtain 0 . Find (i) the probability of getting this result, and (ii) the normalized post-collapse state. (4 marks) (c) Suppose we perform a H B measurement, and obtain q . Find (i) the probability of getting this result, and (ii) the normalized post-collapse state. (d) Find the reduced density operator ρ ^ A = Tr B | ψ ⟩ ⟨ ψ | , expressing your result in (i) bra-ket and (ii) matrix representation. Hence, determine the eigenvalues of ρ ^ A . (5 marks) (e) (Optional) Figure out if A and B are entangled. (0 marks)
Consider a density operator ρ ^ = ∑ λ P λ | λ ⟩ ⟨ λ | where { P λ } is a set of probabilities (i.e., a set of real numbers in [ 0 , 1 ] that sum to 1 ), and { | λ ⟩ } is a set of a normalized states that are not necsarily orthogonal to each other. (a) Explain why each eigenvalue of ρ ^ must be real. (1 mark) (b) Let | n ⟩ be an eigenstate. Prove that its eigenvalue p n is non-negative. Hint: consider Tr ( ρ ^ | n ⟩ ⟨ n | ) . (4 marks) (c) Hence, show that the eigenvalues of ρ ^ form a set of probabilities. (1 mark)