xxix Symbols mixed density matrix density matrix for the final state of a system g density matrix for the initial state of a system reduced density matrix of the j-th subsystem PSA density matrix of the system plus apparatus DSAE density matrix of the system plus apparatus plus environment 2=2)- variance of O standard deviation (square root of the variance) of元 哈=)-月 variance ofp standard deviation(square root of the variance) of px 6+=le (gl raising operator 6-=1g (el lowering operator =(6x,Gy,) Pauli(two-dimensional)spin matrices wave component of the spin ket of the object system time interval,interaction time between two or more systems 产1() decoherence time angle,spherical coordinate angle operator eigenket of the angle operator 1p）.p1 state vectors p传) eigenfunctions of the observable with eigenvector) 9) plane waves c spherical waves Pp(x) momentum eigenfunctions in the position representation po（x) position eigenfunctions in the position representation Ppo(Px) momentum eigenfunctions in the momentum representation Px(px) position eigenfunctions in the momentum representa E(x) scalar product(x) 4p(E) scalar product(传|nl flux of electric current magnetic flux generic ket for compound systems X()=∫dFx)e classical characteristic function of a random variablesxxix Symbols ˆ ρ˜ mixed density matrix ρˆf density matrix for the final state of a system ρˆi density matrix for the initial state of a system !ˆ j reduced density matrix of the j-th subsystem ρˆSA density matrix of the system plus apparatus ρˆSAE density matrix of the system plus apparatus plus environment σ2 x =  xˆ2  −  xˆ 2 variance of xˆ σx standard deviation (square root of the variance) of xˆ σ2 p =  pˆ2 x  −  pˆx 2 variance of pˆx σp standard deviation (square root of the variance) of pˆx σˆ+ = |e g | raising operator σˆ− = |g e | lowering operator σˆ = (σˆx , σˆ y , σˆz) Pauli (two-dimensional) spin matrices ς(s) wave component of the spin |ς ket of the object system τ time interval, interaction time between two or more systems τd  γ −1 λT x 2 decoherence time φ angle, spherical coordinate φˆ angle operator |φ eigenket of the angle operator |ϕ,  ϕ  state vectors ϕ(ξ ) eigenfunctions of the observable with eigenvector | ξ ϕk (x) plane waves ϕk(r) spherical waves ϕp(x) momentum eigenfunctions in the position representation ϕx0 (x) position eigenfunctions in the position representation ϕ˜ p0 (px ) momentum eigenfunctions in the momentum representation ϕ˜x (px ) position eigenfunctions in the momentum representation ϕξ (x) scalar product x | ξ ϕη(ξ ) scalar product ξ | η % flux of electric current %M magnetic flux |% generic ket for compound systems χξ (η) =  dF(x)eıηx classical characteristic function of a random variable ξ