The Dynamical Additivity and the Strong Dynamical Additivity of Quantum Operations

In this paper, we always assume that H is an N-dimensional complex Hilbert space. Let L(H) be the set of all linear operators from H to H . A state ρ of some quantum system, described by H , is a positive semi-definite operator of trace one, in particular, for each unit vector |ψ〉 ∈ H , the operator ρ = |ψ〉〈ψ| is said to be a pure state. The set of all states onH is denoted by D(H). If X,Y ∈ L(H), then 〈X,Y〉 = Tr(X†Y) defines an inner product on L(H), which is called the Hilbert-Schmidt inner product. The following fact is often used: If X,Y ∈ L(H) are two positive semi-definite operators, it follows from 〈X,Y〉 = Tr(X†Y) = Tr(XY) = Tr(X 1 2 YX 1 2 ) that 〈X,Y〉 = 0 if and only if XY = 0, that is, X and Y are orthogonal if and only if XY = 0. Let S ,T ∈ L(H ⊗ H) be two positive semi-definite operators. Denote S 1 = Tr2(S ), T1 = Tr2(T ), S 2 = Tr1(S ) and T2 = Tr1(T ). Then S 1,T1, S 2,T2 ∈ L(H) are all positive semi-definite operators. If S 1T1 = S 2T2 = 0, then S and T is said to be bi-orthogonal (see [7]). Let {|i〉} be the standard basis ofH . For each P = i, j pi j|i〉〈 j| ∈ L(H), we denote vec(P) = ∑ i j pi j|i j〉, then vec defined a linear map from L(H) to H ⊗ H . Moreover, if HA and HB are two Hilbert spaces, {|m〉} and {|μ〉} are their standard bases, respectively, then we can also define a map vec that describes a change of the standard basis from L(HA ⊗ HB) to HA ⊗ HA ⊗ HB ⊗ HB, that is, vec(|m〉〈n| ⊗ |μ〉〈ν|) = |mn〉 ⊗ |μν〉. Moreover, if X ∈ L(HA), Z ∈ L(HB), then vec(X ⊗ Z) = vec(X) ⊗ vec(Z) (see [14]).