Postulates on mixed states Unitary operation U: PHUpU For pure state p=lo)(l it becomes UpU U|p)(|Ut=|")(φ' Where|φ")=Uld Orthogonal measurement (l 1), a)): outcome i occurs with probability k (yplplap)12, and the system collapses to p′=|y2)(y For pure state p=|〉φh, the probability is‖〈φlψi)}2, and the collapsed state is ly il If we measure p E caxa in the computational basis (1),2),,d), then Prloutcome i occurs= p the i-th diagonal entry ofPostulates on mixed states • Unitary operation 𝑈: 𝜌 ↦ 𝑈𝜌𝑈 † – For pure state 𝜌 = 𝜙 𝜙 , it becomes 𝑈𝜌𝑈 † = 𝑈 𝜙 𝜙 𝑈 † = 𝜙 ′ 𝜙 ′ where 𝜙 ′ = 𝑈 𝜙 . • Orthogonal measurement 𝜓1 , … 𝜓𝑑 : outcome 𝑖 occurs with probability 𝜓 𝜌 𝜓 2 , and the system collapses to 𝜌 ′ = 𝜓𝑖 𝜓𝑖 . – For pure state 𝜌 = 𝜙 𝜙 , the probability is 𝜙|𝜓𝑖 2 , and the collapsed state is 𝜓𝑖 . • If we measure 𝜌 ∈ ℂ 𝑑×𝑑 in the computational basis 1 , 2 , … , 𝑑 , then Pr outcome 𝑖 occurs = 𝜌𝑖𝑖, the 𝑖-th diagonal entry of 𝜌