Postulates of Quantum Mechanics Postulates 3:Quantum measurements are described by a collection of measurement operators {Mmsms.If the state is),then p(m)=Prob[outputs m]=(MT Mm) The state of the system after the measurement is Mmlp） MMml) The measurement operators satisfy Mim Mm =1 m=1
Postulates of Quantum Mechanics • Postulates 3: Quantum measurements are described by a collection of measurement operators 𝑀𝑚 1≤𝑚≤𝑛 . If the state is |𝜓⟩, then 𝑝 𝑚 =Prob[outputs m]=⟨𝜓 𝑀𝑚 † 𝑀𝑚 𝜓⟩ The state of the system after the measurement is 𝑀𝑚 𝜓 ⟨𝜓 𝑀𝑚 † 𝑀𝑚 𝜓⟩ The measurement operators satisfy 𝑚=1 𝑛 𝑀𝑚 † 𝑀𝑚 = 𝐼
Postulates of Quantum Mechanics Postulates 3':Quantum measurements are described by a collection of measurement operators{Mmsms If the state isp,then p(m)=Prob[outputs m]=TrM Mmp The state of the system after the measurement is MmPMn TrMmPMm The measurement operators satisfy n Mih Mm =I m=1
Postulates of Quantum Mechanics • Postulates 3’: Quantum measurements are described by a collection of measurement operators 𝑀𝑚 1≤𝑚≤𝑛 . If the state is 𝜌, then 𝑝 𝑚 =Prob[outputs m]=𝑇𝑟𝑀𝑚 † 𝑀𝑚𝜌 The state of the system after the measurement is 𝑀𝑚𝜌𝑀𝑚 † 𝑇𝑟𝑀𝑚𝜌𝑀𝑚 † The measurement operators satisfy 𝑚=1 𝑛 𝑀𝑚 † 𝑀𝑚 = 𝐼
Postulates of Quantum Mechanics Postulates 4:Given systems 1,...,n,if the system i is in state i),then the composite system is in state1〉☒…☒lpn). Postulates 4':Given systems 1,...,n,if the system i is in state pi,then the composite system is in state p1☒…&pn:
Postulates of Quantum Mechanics • Postulates 4: Given systems 1, … , 𝑛, if the system 𝑖 is in state |𝜓𝑖 ⟩, then the composite system is in state 𝜓1 ⊗ ⋯ ⊗ 𝜓𝑛 . • Postulates 4’: Given systems 1, … , 𝑛, if the system 𝑖 is in state 𝜌𝑖 , then the composite system is in state 𝜌1 ⊗ ⋯ ⊗ 𝜌𝑛
Classical states ·0→10〉1→11) x=x1.xn∈{0,1n→lx1,…,xn》 p is a classical distribution over {1,2,....,n}.Its density operator is 三 … It is a diagonal matrix
Classical states • 0 → |0⟩ 1 → |1⟩ 𝑥 = 𝑥1 … 𝑥𝑛 ∈ 0,1 𝑛 → 𝑥1 ,… , 𝑥𝑛 • 𝑝 is a classical distribution over 1,2, … , 𝑛 . Its density operator is 𝜌 = 𝑝1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑝𝑛 It is a diagonal matrix
Classical gates Given a one-to-one f:{0,13{0,1)m 0 : 0 x0.…010…0 0 0 {0,1n×{0,1}n f(x) U is a permutation matrix.UTU=I Ulx)=If(x)》
Classical gates Given a one-to-one 𝑓: 0,1 𝑛 → 0,1 𝑛 𝑈𝑓 = 𝑥 0 ⋮ 0 0 ⋯ ⋯ 0 1 0 ⋯ 0 0 ⋮ 0 0,1 𝑛× 0,1 𝑛 𝑓(𝑥) 𝑈𝑓 is a permutation matrix. 𝑈𝑓 †𝑈𝑓 = 𝐼 𝑈𝑓 𝑥 = |𝑓 𝑥 ⟩