Finite-bath dephasing revival violates semigroup

Let $Z=\mathrm{diag}(1,-1)$, take $g>0$ in ${\mathrm{s}}^{-1}$, and set $H=\mathrm{\hslash }gZ\otimes Z$. With ${\sigma }_E=I_2\mathrm{/}2=\frac{1}{2}{\sum }_{b=\pm 1}\mathrm{\mid }b⟩⟨b\mathrm{\mid }$ and $U_t=e^{-itH\mathrm{/}\mathrm{\hslash }}$, the power-series definition of the finite operator exponential in [F3] and the $Z$-eigenbasis give $U_t={\sum }_{b=\pm 1}\mathrm{diag}(\exp (-ibgt),\exp (ibgt))\otimes \mathrm{\mid }b⟩⟨b\mathrm{\mid }$. Applying [F1] to the two orthogonal environment sectors yields ${\mathrm{\Phi }}_t(\rho )=\frac{1}{2}(V_t\rho V_t^{\ast }+V_t^{\ast }\rho V_t)$, where $V_t=\mathrm{diag}(\exp (-igt),\exp (igt))$. In matrix coordinates, ${\mathrm{\Phi }}_t\text{\hspace{0.17em}⁣}\left(\begin{array}{cc}a& x\\ y& d\end{array}\right)=\left(\begin{array}{cc}a& \kappa (t)x\\ \kappa (t)y& d\end{array}\right)$ with $\kappa (t):=\frac{1}{2}(\exp (-2igt)+\exp (2igt))$. By [F3], each ${\mathrm{\Phi }}_t$ is a channel.

By [F7], choose a dimensionless $\theta \in [0,2\pi )$ with $(\cos \theta ,\sin \theta )=(0,1)$; hence $\theta >0$. Equation [F6] gives $\exp (i\theta )=i$, and [F5] with [F4] gives $\exp (-i\theta )=-i$ and $\exp (2i\theta )=\exp (-2i\theta )=-1$. Set $\tau =\theta \mathrm{/}(2g)>0$ and ${\rho }_{+}=\mathrm{\mid }+⟩⟨+\mathrm{\mid }$, with $\mathrm{\mid }+⟩=(\mathrm{\mid }0⟩+\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$. By step 1.1, $\kappa (\tau )=\frac{1}{2}(-i+i)=0$ and $\kappa (2\tau )=\frac{1}{2}(-1-1)=-1$. Therefore ${\mathrm{\Phi }}_{\tau }({\mathrm{\Phi }}_{\tau }({\rho }_{+}))=I_2\mathrm{/}2$ whereas ${\mathrm{\Phi }}_{2\tau }({\rho }_{+})=\mathrm{\mid }-⟩⟨-\mathrm{\mid }$, where $\mathrm{\mid }-⟩=(\mathrm{\mid }0⟩-\mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$. The outputs differ, so ${\mathrm{\Phi }}_{2\tau }\mathrm{\ne }{\mathrm{\Phi }}_{\tau }\circ {\mathrm{\Phi }}_{\tau }$; the family is not a semigroup.