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.



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