Theorem: Exact retained-source channel

For input Bloch vector $(u_x,u_y,u_z)$, the retained source has Bloch vector

Equivalently,

For every finite $a_1$, $q>0$.

Proof. Write $A=X(a_1)$, $B=X(a_2)$. The second pulse leaves $X$ invariant, while the first gives

Averaging in the centered Gaussian state gives $qX$.

For $Y$,

In a diagonal $Z$-block, the first pulse shifts $B$ by $-z\theta$. The diagonal expectation is therefore $r\sin (2\theta )$, independent of $z$. In an off-diagonal block use

The cosine sandwich contributes $c_{+}Y$.

Similarly,

Its diagonal average is $r\cos (2\theta )Z$, and the sine sandwich gives $c_{-}X$. This proves the adjoint identities and all displayed formulas.