 ${\mathbb{P}}(X_0=x_0,\ldots,X_n=x_n) = {\operatorname{tr}}t^{(n)}_{x_n} \cdots t^{(0)}_{x_0}\, \rho \,     {t^{(0)}_{x_0}}^* \cdots {t^{(n)}_{x_n}}^*$ is independent of the input state $\rho$.  When these conditions hold, then 