  The basic notions of quantum mechanics are formulated in terms of separable
infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert
lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions
of state and observable can be formulated as kinds of measures as in [21]. The
aim of this paper is to show that there is a good notion of computability for
these data structures in the sense of Weihrauch's Type Two Effectivity (TTE)
[26].
  Instead of explicitly exhibiting admissible representations for the data
types under consideration we show that they do live within the category
$\mathbf{QCB}_0$ which is equivalent to the category $\mathbf{AdmRep}$ of
admissible representations and continuously realizable maps between them. For
this purpose in case of observables we have to replace measures by valuations
which allows us to prove an effective version of von Neumann's Spectral
Theorem.
