Proposition: Pilot implementation stability

Compare two active instruments with parameters $\eta ,\tilde{\eta }\in [0,{\eta }_{\ast }]$, where ${\eta }_{\ast }<1$, and feedback phases ${\vartheta }_h,{\tilde{\vartheta }}_h$. For the same routing table,

The bound may be capped by $2$.

Proof. The derivatives of the scalar entries of $M_a$ have magnitude at most $1/(2\sqrt{2(1-{\eta }_{\ast })})$. Hence the first term bounds $\Vert M_a-{\tilde{M}}_a{\Vert }_{\infty }$. Also

The branch operators $K_a=U_a{M}_a$ are contractions, and

This follows by expanding $K_{a}X{K}_a^{†}-{\tilde{K}}_{a}X{\widetilde{K_a}}^{†}$ into two terms, also after amplification. Summing the two branch bounds and using Proposition prop:diamond-column gives (eq:pilot-stability). Two channels have diamond distance at most $2$.