Continuous finite GKSL channel propagators

Let $\mathcal{E}={\mathrm{End}}_{\mathbb{C}}(M_d(\mathbb{C}))$ with the induced map norm. It has finite complex dimension $d^4$, so it is complete as a real normed space by [F5]. Set $M:={\sup }_{u\in I}\mathrm{\parallel }{\mathcal{L}}_u\mathrm{\parallel }<\mathrm{\infty }$. For fixed $s$, define $D_0(t)=I$ and recursively $D_n(t)={\int }_s^t{\mathcal{L}}_u{D}_{n-1}(u)du$ for $n\ge 1$, using [F6]. Induction with the integral norm bound gives $\mathrm{\parallel }{D}_n(t)\mathrm{\parallel }\le M^n(t-s{)}^n\mathrm{/}n!$; hence ${\sum }_{n=0}^{\mathrm{\infty }}{D}_n(t)$ converges uniformly on $[s,s_1]$. Calling its sum $\mathrm{\Lambda }(t,s)$, uniform convergence and boundedness of ${\mathcal{L}}_u$ allow passage through the integral, giving $\mathrm{\Lambda }(t,s)=I+{\int }_s^t{\mathcal{L}}_u\mathrm{\Lambda }(u,s)du$. Also $\mathrm{\parallel }\mathrm{\Lambda }(u,s)\mathrm{\parallel }\le e^{M(u-s)}$.

Let $t_j=s+j(t-s)\mathrm{/}n$ be a uniform partition of $[s,t]$ and freeze ${\mathcal{L}}_u$ to ${\mathcal{L}}_{t_j}$ on each interval $[t_j,t_{j+1})$. The resulting propagator is the time-ordered composition $e^{(t-t_{n-1}){\mathcal{L}}_{t_{n-1}}}\circ \cdots \circ e^{(t_1-t_0){\mathcal{L}}_{t_0}}$. Each factor is a channel by [F2], and the composition is a channel by [F3].

By the fundamental theorem [F6], the integral equation in step 1.1 is differentiable and satisfies ${\mathrm{\partial }}_t\mathrm{\Lambda }(t,s)={\mathcal{L}}_t\circ \mathrm{\Lambda }(t,s)$, with $\mathrm{\Lambda }(s,s)=I$. If another solution differs by $D$, then $D(t)={\int }_s^t{\mathcal{L}}_{u}D(u)du$. With $C={\sup }_{u\in [s,s_1]}\mathrm{\parallel }D(u)\mathrm{\parallel }<\mathrm{\infty }$, repeated application of the integral norm bound gives $\mathrm{\parallel }D(t)\mathrm{\parallel }\le C{M}^n(t-s{)}^n\mathrm{/}n!$ for every $n\ge 0$. This tends to zero, proving uniqueness.

Let ${\delta }_n={\sup }_{u\in [s,t]}\mathrm{\parallel }{\mathcal{L}}_u^{(n)}-{\mathcal{L}}_u\mathrm{\parallel }$; uniform continuity gives ${\delta }_n\to 0$. Subtract the integral equations for the frozen propagator ${\mathrm{\Lambda }}_n$ and $\mathrm{\Lambda }$. For $s\le u\le t$, the norm difference is bounded by $M{\int }_s^u\mathrm{\parallel }{\mathrm{\Lambda }}_n(v,s)-\mathrm{\Lambda }(v,s)\mathrm{\parallel }dv+{\delta }_n(t-s)e^{M(t-s)}$, using the bound in step 1.1. Iterating this scalar integral inequality yields $\mathrm{\parallel }{\mathrm{\Lambda }}_n(t,s)-\mathrm{\Lambda }(t,s)\mathrm{\parallel }\le {\delta }_n(t-s)e^{2M(t-s)}\to 0$. The channel set is closed under this norm limit by [F3], so $\mathrm{\Lambda }(t,s)$ is a channel. For $s=t$ it is $I$, also a channel.