U(t)) \vert \Sigma \rangle.$$ For notational convenience we write $$w_{jk} = \alpha_j^\ast \alpha_k  \label{w_jk}$$ and note that $w = \sum_{jk} w_{jk}  \vert e_j \rangle \langle e_k \vert$ is a density matrix for $\mathbb{C}^2
$.

The expectation with respect to the superposition state $\vert \Sigma \rangle$ is given by $$\tilde \mu_t( A \otimes X) = \mathbb{E}_{\psi} [ A \otimes X(t) ] = \langle \Sigma \vert
(A \otimes X(t)) \vert \Sigma \rangle = \sum_{jk} w_{jk} a_{jk} \mu^{jk}_t(X).
\label{eq:master-mu-rep1}$$ This expectation is correctly normalized, $\mu_t(I \otimes I)=1$, and the expectations $\mu^{jk}_t(X)$ defined in Section \[sec:master\] are scaled components of $\tilde \mu_t(A\otimes X)$: $$\mu^{jk}_t(X) = \frac{ \tilde \mu_t( \vert e_j \rangle \langle e_k \vert  \otimes X)}{ w_{jk} },
\label{eq:master-mu-rep2}$$ for $w_{jk} \neq 0$, otherwise it can be set to, say, 0. We also have $$\mu^{jk}_t(X) = \frac{ w_{11}  \tilde\mu_t( \vert e_j \rangle \langle e_k \vert  \otimes X) }{ w_{jk}  \tilde \mu_t(  \vert e_1 \rangle \langle e_1 \vert \otimes I) } .
\label{eqmui-jk-bayes}$$

Note that in the extended space the Schrödinger and Heisenberg pictures are related by $$\mathbb{E}_{\Sigma(t) } [ A \otimes X \otimes F ] = \mathbb{E}_{\Sigma} [ A
\otimes U^\ast(t) (X \otimes F) U(t) ] .$$

In order to derive the equation for 