Editing Quantum decoherence (section)

===Dirac notation===
Using [[Dirac notation]], let the system initially be in the state

: <math>|\psi\rang = \sum_i |i\rang \lang i |\psi\rang,</math>

where the <math>|i\rang</math>s form an einselected [[eigenbasis|basis]] (''environmentally induced selected eigenbasis''<ref name="zurek03"/>), and let the environment initially be in the state <math>|\epsilon\rang</math>. The [[vector basis]] of the combination of the system and the environment consists of the [[tensor product]]s of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as

: <math>|\text{before}\rang = \sum_i |i\rang |\epsilon\rang \lang i|\psi\rang,</math>

where <math>|i\rang |\epsilon\rang</math> is shorthand for the tensor product <math>|i\rang \otimes |\epsilon\rang</math>. There are two extremes in the way the system can interact with its environment: either (1) the system loses its distinct identity and merges with the environment (e.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is disturbed (e.g. the idealized non-disturbing measurement). In general, an interaction is a mixture of these two extremes that we examine.

====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"/>

====System not disturbed by environment====
In an idealized measurement, the system disturbs the environment, but is itself undisturbed by the environment. In this case, each element of the basis interacts with the environment such that

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

and so

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

In this case, [[unitarity (physics)|unitarity]] demands that

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

where <math>\lang \epsilon_i | \epsilon_i \rang = 1</math> was used. ''Additionally'', decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that

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

As before, this is the defining characteristic for decoherence to become einselection.<ref name="zurek03"/> The approximation becomes more exact as the number of environmental degrees of freedom affected increases.

Note that if the system basis <math>|i\rang</math> were not an einselected basis, then the last condition is trivial, since the disturbed environment is not a function of <math>i</math>, and we have the trivial disturbed environment basis <math>|\epsilon_j\rang = |\epsilon'\rang</math>. This would correspond to the system basis being degenerate with respect to the environmentally defined measurement observable. For a complex environmental interaction (which would be expected for a typical macroscale interaction) a non-einselected basis would be hard to define.