Understanding interference and decoherence

[Talisman]

I'm trying to make sense of various explanations of why decoherence causes interference to disappear, but I'm afraid I don't quite grok it.

There's a bit of explanation in this thread.

And some more on wikipedia.

The first link starts with |\psi{\rangle} = |a{\rangle} + |b{\rangle} and considers {\langle}\psi | \psi{\rangle} = {\langle}a|a{\rangle} + {\langle}b|b{\rangle} + 2 Re({\langle}a|b{\rangle}). Clearly a and b are not meant to be basis vectors here; in that case, why did we write \psi as a sum of them to begin with?

Wikipedia shows the following:
prob(\psi \Rightarrow \phi) = |{\langle} \psi |\phi {\rangle}|^2 = |\sum_i\psi^*_i \phi_i |^2 = \sum_{ij} \psi^*_i \psi_j \phi^*_j\phi_i= \sum_{i} |\psi_i|^2|\phi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i
In both cases, when we work out the algebra in the inner product, we find that the expansion contains terms involving the product of components of each state, which are called "cross terms."

What I don't understand is the significance of taking the inner product of a state with itself in the first case, or what the "transition probability" refers to in the second. Also, how does this relate to the double slit experiment? Aren't the position states represented by the electron going through each slit orthogonal?

 

[azntoon]

The inner product of a state with itself in the first case is a measure of the probability that the system will remain in its original state, while the transition probability in the second case is the probability that the system will transition from its initial state to its final state, given the two states. This can be seen as relating to the double slit experiment in that the initial state is the electron going through one of the two slits, and the final state is the electron going through both slits at the same time. As the two states are orthogonal, the transition probability between them is zero, and so interference is not observed. This is because the cross terms between the two states cancel each other out, meaning that the probability of the electron being detected at any given point on the screen is not affected by the electron going through both slits (i.e., there is no interference). Decoherence prevents interference from occurring by introducing random noise into the system which causes the two states to become entangled with their environment, making them no longer orthogonal. This means that the cross terms between the two states no longer cancel each other out, and so the interference pattern can be observed.