Quatum computing two-part problems. i) Consider the two states
|ψ ⟩ =
1
√
2
(|000⟩ + |111⟩) (0.1)
|ϕ⟩ =
1
√
3
(|100⟩ + |010⟩ + |001⟩) (0.2)
Calculate the density matrix ρ for these states and then calculate Tr23(ρ) (partial
trace by over the 2nd and 3rd qubit). Are any of these states maximally mixed
states?
ii) Consider the 4 × 4 matrix (density matrix)
|u⟩⟨u| = (uju∗
k) , j, k = 1, . . . , 4
in the product Hilbert space HA ⊗ HB ≡ C4, where HA = HB = C2. (i) Calculate
trA(|u⟩⟨u|).