# Mathematical aspects of decoherence

I am trying to understand the maths behind decoherence. I first state the framework I am working with, and then I ask my questions.

Let $$H_A$$, $$H_B$$ be finite dimensional Hilbert spaces ("the system" and "the observer") and let $$H_C := H_A \otimes H_B$$. Let $$(\phi_i)_i$$ be an orthonormal basis of $$H_A$$, and $$\psi \in H_B$$. Let $$H$$ be a self-adjoint operator on $$H_C$$ and let $$(a_i)_i$$ be complex numbers such that $$\sum \vert a_i \vert^2 = 1$$.

Let $$U(t) := e^{-iHt}$$. Then let $$\eta_{ij}(t)$$ be such that for all $$t$$, $$i$$, $$U(t)(\phi_i \otimes \psi) = \sum_j \phi_j \otimes \eta_{ij}(t)$$, so that for all $$t$$, $$U(t)(\sum_i a_i \phi_i \otimes \psi) = \sum_i \sum_j \phi_j \otimes a_i\eta_{ij}(t)$$.

Questions :

1) Decoherence is the name of the fact that $$\sum_{l,m} \overline{a_l}a_m\langle \eta_{li}(t),\eta_{mj}(t)\rangle$$ is "small" whenever $$i \neq j$$. Am I wrong ?

2) What does "small" precisely mean ? Does it mean that $$\sum_{l,m} \overline{a_l}a_m\langle \eta_{li}(t),\eta_{mj}(t)\rangle \to 0$$ ? Does it mean that for any large $$t$$, $$\vert \sum_{l,m} \overline{a_l}a_m\langle \eta_{li}(t),\eta_{mj}(t)\rangle \vert$$ is quite small, and gets smaller when the dimension of $$H_B$$ grows ? Does it mean something else ?

3) Are there theorems that state something like "for all $$H$$ in some class, decoherence happens ?" where decoherence has one of the meanings above ?

EDIT: I changed notations, because $$H$$ as both the Hilbert space and the hamiltonian.