# Predictability in decoherence theory to find the classical states: at which time must we evaluate?

I have read Decoherence, einselection, and the quantum origins of the classical, end a way to quantify the classicality of states is the following.

We have the system $$S$$ and its environment $$E$$. The total Hamiltonian can formally be written: $$H=H_S+H_E+H_{SE}$$.

As far as I understood, the pointer states, which correspond to the state we will observe classically, are ideally states of the system $$S$$ that verify: $$[|\psi_S\rangle \langle \psi_S|,H]=0$$.

However in practice this condition is too stringent and it is not always possible to have such states. Thus, a more general way to define the classical states is to find the ones that will remain the more pure after interaction.

To do this, we define the "measure of predictability" as being: $$s_{\psi}(t)=Tr(\left(\rho_{|\psi_S \rangle}(t)\right)^2)$$ where $$\rho_{|\psi_S \rangle}(t)=|\psi_S\rangle \langle \psi_S|$$. Basically it is the purity of the state $$|\psi_S \rangle$$ after it has interacted with the environment for a time $$t$$. The classical states would correspond to the one that will have the bigger purity after interaction.

However, which time must we take to evaluate this function ? Indeed this measure is "time dependant". How can I know then which state are the most classical: the answer shouldn't depend on $$t$$.