Quantum mechanics in one slide PhysIcs Math a|0)+611 Physical System (>Unit Vector Evolution台 Unitary Matrixⅸ Measurement Projection A quantum bit Composition Tensor Product (qubit Classical State space for 2 bits 0 0 combinations 00, 01, 10, 11) Quantum State space for 2 qubits space span{00),|01),110),11)
Quantum mechanics in one slide Physics Math Composition Tensor Product Measurement Projection Evolution Unitary Matrix Physical System Unit Vector 1 0 1 0 Classical: Quantum: State space for 2 bits: combinations 00,01,10,11 State space for 2 qubits: space span 00 , 01 , 10 , 11 A quantum bit (qubit) 𝛼 0 + 𝛽 1 0 1 𝛽 𝛼 𝛼 1 1 0 0
Density matrix If a system is in stateψ1〉 with probability"1, and in state 1 2) with probability p2, then the system is in a mixed state The mixed state is represented as a density matrix p=p1|y)y1|+p21yb2)(y2 In general, if a system is in state l i> with probability pi, then the mixed state is p=∑p21vz For pure state ) p=lyb bl
Density matrix • If a system is in state 𝜓1 with probability 𝑝1, and in state 𝜓2 with probability 𝑝2, then the system is in a mixed state. • The mixed state is represented as a density matrix 𝜌 = 𝑝1 𝜓1 𝜓1 + 𝑝2 𝜓2 𝜓2 . • In general, if a system is in state 𝜓𝑖 with probability 𝑝𝑖 , then the mixed state is 𝜌 = σ𝑖 𝑝𝑖 𝜓𝑖 𝜓𝑖 • For pure state 𝜓 , 𝜌 = 𝜓 𝜓
Density matriX Fact. A matrix p is a density matrix of some mixed quantum state iff p is positive semi-definite(psd) Tr(p=1 Recall Amatrix M is psd if all its eigenvalues are nonnegative. Equivalently, if (v Mv)20,Vv The trace of a matrix M is Tr(M)=2i Mi
Density matrix • Fact. A matrix 𝜌 is a density matrix of some mixed quantum state iff – 𝜌 is positive semi-definite (psd) – Tr 𝜌 = 1. • Recall: – A matrix 𝑀 is psd if all its eigenvalues are nonnegative. Equivalently, if 𝑣 𝑀 𝑣 ≥ 0, ∀𝑣. – The trace of a matrix 𝑀 is Tr 𝑀 = σ𝑖 𝑀𝑖𝑖
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 of
Postulates 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
Composition of p, and p2 is just P1 0 p2 For pure state p1=|中1)(中1|andp2= 2〉(φ2, the joint state is p1)(φ1∞|φ2)(φ2 =(|1)|φ2)(1∞(2|) Recall tensor product of matrices 1b1a1b12 12011a12b C11012 12 11021110221202101222 21a22 21 22 21b1a21b12a22b11 a21b21a21b22a22b21a22b22
• Composition of 𝜌1 and 𝜌2 is just 𝜌1 ⊗ 𝜌2 – For pure state 𝜌1 = 𝜙1 𝜙1 and 𝜌2 = 𝜙2 𝜙2 , the joint state is 𝜙1 𝜙1 ⊗ 𝜙2 𝜙2 = 𝜙1 ⊗ 𝜙2 𝜙1 ⊗ 𝜙2 • Recall tensor product of matrices: 𝑎11 𝑎12 𝑎21 𝑎22 ⊗ 𝑏11 𝑏12 𝑏21 𝑏22 = 𝑎11𝑏11 𝑎11𝑏12 𝑎11𝑏21 𝑎11𝑏22 𝑎12𝑏11 𝑎12𝑏12 𝑎12𝑏21 𝑎12𝑏22 𝑎21𝑏11 𝑎21𝑏12 𝑎21𝑏21 𝑎21𝑏22 𝑎22𝑏11 𝑎22𝑏12 𝑎22𝑏21 𝑎22𝑏22