Postulate 2: operation Evolution: The evolution of a closed quantum system is described by a unitary transformation That is, if a system is in state 11 at time ti and in state l 2 at time t2 then there is a unitary transformation U s t ly2)=U/y1) Unitary transformation Ut=u-1 Recall: Ut=(U)*, transpose complex conjugate You can think of it as a rotation operation
Postulate 2: operation • Evolution: The evolution of a closed quantum system is described by a unitary transformation. • That is, if a system is in state 𝜓1 at time 𝑡1, and in state 𝜓2 at time 𝑡2, then there is a unitary transformation 𝑈 s.t. 𝜓2 = 𝑈 𝜓1 . • Unitary transformation: 𝑈 † = 𝑈 −1 – Recall: 𝑈 † = 𝑈 𝑇 ∗ , transpose + complex conjugate – You can think of it as a rotation operation
Postulate 3: measurement Measurement: We can only observe a quantum system by measuring it The outcome of the measurement is random And the system is changed by the measuremen
Postulate 3: measurement • Measurement: We can only observe a quantum system by measuring it. • The outcome of the measurement is random. • And the system is changed by the measurement
If we measure qubit a[0)+B 1 in the computational basis 10),|1), then outcome“0” 0)+B|1) occurs with prob lal, and outcome 1"occurs with prob 1B12 The system becomes 0)if outcome“ 0 occurs,and A quantum bit becomes|l) f outcome“1” (qubit occurs The system collapses
• If we measure qubit 𝛼 0 + 𝛽 1 in the computational basis 0 , 1 , then outcome “0” occurs with prob. 𝛼 2 , and outcome “1” occurs with prob. 𝛽 2 . • The system becomes 0 if outcome “0” occurs, and becomes 1 if outcome “1” occurs. – The system collapses. A quantum bit (qubit) 𝛼 0 + 𝛽 1 0 1
Measurement on general states In general, an orthogonal measurement of a d-dim state is given by an orthonormal bass{ψ1)…ya) If we measure state )in basis {y1)…ybd), then outcome i∈{1,…,l occurs with prob. I(lvill2 The system collapses to l i if outcome i occurs
Measurement on general states • In general, an orthogonal measurement of a 𝑑-dim state is given by an orthonormal basis 𝜓1 ,… 𝜓𝑑 . • If we measure state 𝜙 in basis 𝜓1 ,… 𝜓𝑑 , then outcome 𝑖 ∈ 1,… , 𝑑 occurs with prob. 𝜙|𝜓𝑖 2 . • The system collapses to 𝜓𝑖 if outcome 𝑖 occurs
Postulate 4: composition Composition: The state of the joint system (S1, S2, where S, is in state l 1 and s2 in y2),is|y1)②|v2) tensor product of vectors (a1,a2)⑧(b1,b2b3)=(a1b1,a1b2,a1b3,a2b1,a2b2a2b3) dim( 1&1 2))=dim( 1)). dimdya2)) size (l idly b2))=size (ly1))+ size(l b2)) size: number of qubits Notation:|0)8n=|0)⑧…|0), m times
Postulate 4: composition • Composition: The state of the joint system (𝑆1, 𝑆2), where 𝑆1 is in state 𝜓1 and 𝑆2 in 𝜓2 , is 𝜓1 ⊗ 𝜓2 . • ⊗: tensor product of vectors. – 𝑎1, 𝑎2 ⊗ 𝑏1, 𝑏2, 𝑏3 = (𝑎1𝑏1, 𝑎1𝑏2, 𝑎1𝑏3, 𝑎2𝑏1, 𝑎2𝑏2, 𝑎2𝑏3). – dim 𝜓1 ⊗ 𝜓2 = dim 𝜓1 ⋅ dim 𝜓2 – size 𝜓1 ⊗ 𝜓2 = size 𝜓1 + size 𝜓2 • size: number of qubits. • Notation: 0 ⊗𝑛 = 0 ⊗ ⋯ ⊗ 0 , 𝑛 times