Exact objects and state approximation

Let ${\mathcal{H}}_{\mathrm{osc}}=L^2(\mathbb{R})$ have oscillator number basis $|n⟩$, $n\ge 0$. In this section $a$ denotes the oscillator annihilation operator, not the two-level matrix of Definition def:model. Let

be multiplication by $x$, with PVM

For an integer $d\ge 1$, define