Editing Quantum decoherence (section)

====System absorbed by environment====
If the environment absorbs the system, each element of the total system's basis interacts with the environment such that

: <math>|i\rang |\epsilon\rang </math> evolves into <math>|\epsilon_i\rang,</math>

and so

: <math>|\text{before}\rang</math> evolves into <math>|\text{after}\rang = \sum_i |\epsilon_i\rang \lang i|\psi\rang.</math>

The [[unitarity]] of time evolution demands that the total state basis remains [[orthonormal]], i.e. the [[scalar product|scalar]] or [[inner product]]s of the basis vectors must vanish, since <math>\lang i|j\rang = \delta_{ij}</math>:

: <math>\lang\epsilon_i|\epsilon_j\rang = \delta_{ij}.</math>

This orthonormality of the environment states is the defining characteristic required for einselection.<ref name="zurek03"/>