Proposition: Finite Hamiltonian perturbation bound

For bounded self-adjoint time-dependent generators $H(t),\tilde{H}(t)$, with the same initial unitary,

Consequently,

Proof. Differentiate $U(t,r)\tilde{U}(r,s)$ and integrate. All surrounding propagators are unitary. For channels, expand $UX{U}^{†}-\tilde{U}X{\tilde{U}}^{†}$ into two terms, also after tensoring with an identity.