By Romano, R
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3 2. 12) and states becomes less ordered as the time passes. For instance, in Chapters 3 and 5 we shall describe irreversible timeevolutions that are entropy increasing. 8, any entropy non decreasing time-evolution can be fully characterized as follows. e. 7) is unital), where u, v and w are the parameters defined ˜ is the matrix in eq. 20). 10) is symmetric [39, 55]. Proof: The eigenvalues, respectively eigenvectors, of a generic initial state represented by the vector |ρ(0) = (1/2, ρ1 , ρ2 , ρ3 ) with |ρ|2 ≡ 3j=1 (ρj )2 1/4, are ρ± = 1 1 (1 ± 2|ρ|); |ρ± = (|ρ|, ±ρ1 , ±ρ2 , ±ρ3 ).
8)). Their expressions are listed in Appendix A, both for stable and unstable systems. 8 For later convenience we observe that λ1 , λ2 , σ, ξ ∈ R and δ1 , δ2 , φ1 , φ2 , ω, θ ∈ C; on the other hand Λ, Ξ, Σ, Ω ∈ R and ∆, Φ, Θ ∈ C. In the basis of the eigenvectors of H and in the chosen vectorial representation, the evolution operator is denoted by Gˆtd . 45) we find ˆd Gˆtd0 = eH t e−γ1 t 0 0 0 0 e−γ2 t 0 0 . 32), as Gˆtd1 = t 0 ˆ d (t−s) eH ˆ d eHˆ d s . 4. PERTURBATIVE EXPANSION 43 The operator Gˆtd so obtained contains terms which are linear in t; they disappear if we reconstruct the exponential dependences out of them.
32 Chapter 2 Bidimensional Systems In view of later applications, in this chapter we restrict our attention to quantum open systems described by two-dimensional Hilbert spaces. The parameters describing general Markovian dynamics are introduced; positivity and complete positivity of the timeevolution are expressed by sets of inequalities among them. Although the equations of motion can be integrated exactly, in many cases it is preferable to express their solutions as perturbative expansions in the dissipative parameters, provided they are small with respect to the parameters describing the dynamics of the system alone, in absence of the environment.