0 \\
1
\end{array}
\right], \ \ \vert e_1 \rangle  = \left[
\begin{array}{c}
1 \\
0
\end{array}
\right],$$ and let $A$ be an operator acting on $\mathbb{C}^2$, i.e. a complex $2
\times 2$ matrix $$A = \left[
\begin{array}{cc}
a_{11} & a_{10} \\
a_{01} & a_{00}
\end{array}
\right].$$ It may be helpful to think of operators $A \otimes X \otimes F$ on the extended space $\tilde {\mathfrak{H}}$ represented in the Kronecker product form $$A \otimes (X \otimes F) = \left[
\begin{array}{cc}
a_{11} (X \otimes F) & a_{10} (X \otimes F) \\
a_{01} (X \otimes F) & a_{00} (X \otimes F)
\end{array}
\right].$$

We allow the extended system to evolve unitarily according to $I \otimes U(t)
$, where $U(t)$ is the unitary operator for the system and field, given by the Schrödinger equation (\[eq:unitary\]). Note in particular that the system is not coupled to the ancilla $\mathbb{C}^2$, and observables of this two-level system are static. We initialize the extended system in the superposition state $$\vert \Sigma \rangle = \alpha_1 \vert e_1 \eta 1_\xi \rangle + \alpha_0 \vert
e_0 \eta 0 \rangle,  \label{eq:super}$$ where $\vert \alpha_0 \vert^2 + \vert \alpha_1 \vert^2 =1$. This state evolves according to $$\vert \Sigma(t) \rangle = (I \otimes 