I Why does a quantum system with many degrees of freedom imply orthogonality?

[Erland]

Quantum decoherence means that when a quantum system interacts with its environment, coherence is lost, which means that all the density matrix becomes diagonal after the interaction. I never understood why it is so, but I get a clue here:

https://en.wikipedia.org/wiki/Quantum_decoherence#Dirac_notation

It says in particular (where $\epsilon_i$ and $\epsilon_j$ are states of the environment after the interaction):

"Additionally, decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that

"

But why does a "large number of hidden degrees of freedom in the environment" imply approximate orthogonality of the evironmental states?

Is there any mathematical theorem where this is rigorously formulated and proved?

 

[atyy]

https://math.stackexchange.com/ques...s-in-high-dimensional-space-almost-orthogonal

 

[Erland]

That was what I wanted to know. Thanks!