content_image/365981.jpg content_image/365980.jpg content_image/365979.jpg content_image/365978.jpg content_image/365977.jpg content_image/365976.jpg
the eigenstates of H without the spin-environment interaction) gives, for the time-evolved density matrix of the total spin+environment system: 

<img src='content_image/365981.jpg'>

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: 

<img src='content_image/365980.jpg'>

Expanding | ψ 〉 as c$_{+}$ | + 〉 + c$_{−}$ |−〉 , we can rewrite the above expression as 

<img src='content_image/365979.jpg'>

where P , Q , R , T are complex matrices with elements 

<img src='content_image/365978.jpg'>

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$_{lk}$ : 

<img src='content_image/365977.jpg'>

<img src='content_image/365976.jpg'>

