the eigenstates of H without the spin-environment interaction) gives, for the time-evolved density matrix of the total spin+environment system: Further, tracing over the environment states, gives the following reduced system density matrix elements ( ρ$_{s}$ )$_{kl}$ in the basis of spin states | k 〉 and | l 〉 where k, l denote either the plus or minus state: Expanding | ψ 〉 as c$_{+}$ | + 〉 + c$_{−}$ |−〉 , we can rewrite the above expression as where P , Q , R , T are complex matrices with elements Consider the decoherence of the spin state, quantified in terms of the purity [15] S of the spin, where S = Tr( ρ 2 $_{s}$). Given the above results, S can be evaluated via so that, using P ∗ kl = T$_{lk}$ : S = | c$_{+}$ | 4 ∑ k,l ∈ + , − | Q$_{kl}$ | 2 + | c$_{−}$ | 4 ∑ k,l ∈ + , − | R$_{kl}$ | 2 + | c$_{+}$ | $^{2}$| c$_{−}$ | 2 k,l ∈ + , − [ | P$_{kl}$ | 2 + | T$_{kl}$ | 2 + { Q$_{kl}$R ∗ kl + Q ∗ $_{kl}$R$_{kl}$ } ] +2Re [ ∑ k,l ∈ + , − [ c 2 $_{+}$c ∗ 2 $_{−}$P$_{kl}$ P$_{lk}$ + | c$_{+}$ | $^{2}$c ∗ $_{+}$c$_{−}$ [ Q$_{kl}$P ∗ kl + P ∗ $_{lk}$Q ∗ $_{kl}$] + | c$_{−}$ | $^{2}$c ∗ $_{+}$c$_{−}$ [ R$_{kl}$P ∗ kl + P ∗ $_{lk}$R ∗ $_{kl}$] ] ] . (8) S = ∑ k,l ∈ + , $_{−}$〈 k | ρ$_{s}$ | l 〉〈 l | ρ$_{s}$ | k 〉 = ∑ k,l ∈ + , − | ( ρ$_{s}$ )$_{kl}$ | 2 (7) P$_{kl}$ ( t ) = ∑ m,n,γ,γ ′ p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$〈 k, m | γ 〉〈 γ $^{′}$| l, m 〉〈 γ | + , n 〉〈− , n | γ $^{′}$〉 , Q$_{kl}$ ( t ) = m,n,γ,γ ′ p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$〈 k, m | γ 〉〈 γ $^{′}$| l, m 〉〈 γ | + , n 〉〈 + , n | γ $^{′}$〉 , R$_{kl}$ ( t ) = ∑ m,n,γ,γ ′ p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$〈 k, m | γ 〉〈 γ $^{′}$| l, m 〉〈 γ |− , n 〉〈− , n | γ $^{′}$〉 , T$_{kl}$ ( t ) = m,n,γ,γ ′ p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$〈 k, m | γ 〉〈 γ $^{′}$| l, m 〉〈 γ |− , n 〉〈 + , n | γ $^{′}$〉 (6) ( ρ$_{s}$ )$_{kl}$ = | c$_{+}$ | $^{2}$Q$_{kl}$ ( t ) + | c$_{−}$ | $^{2}$R$_{kl}$ ( t ) + c$_{+}$c ∗ $_{−}$P$_{kl}$ ( t ) + c$_{−}$c ∗ $_{+}$T$_{kl}$ ( t ) , (5) ( ρ$_{s}$ )$_{kl}$ = ∑ m,n,γ,γ ′ p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$〈 k, m | γ 〉〈 γ | ψ, n 〉〈 ψ, n | γ $^{′}$〉〈 γ $^{′}$| l, m 〉 . (4) ρ ( t ) = ∑ n p$_{n}$U [ | ψ 〉〈 ψ | ⊗ | n 〉〈 n | ] U † = ∑ γ,γ ′ ,n p$_{n}$e − i ( E$_{γ}$ − E$_{γ}$ ′ ) $^{t}$| γ 〉〈 γ $^{′}$| 〈 γ | ψ, n 〉〈 ψ, n | γ $^{′}$〉 . (3) 5