$$|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle$$ $$\hat{T}|\psi(t)\rangle = |\psi(-t)\rangle$$ $$U(t,t_0)=e^{- \frac{i}{\hbar}\int_{t_0}^{t}\hat{H}(\tau)\,d\tau}$$ $$\rho_{CTC}=\text{Tr}_S\left[U(\rho_S\otimes\rho_{CTC})U^\dagger\right]$$ $$\boxed{ \exists\ \rho_{CTC} \text{ such that } \rho_{CTC}^{out}=\rho_{CTC}^{in} }$$ $$\Rightarrow \text{Closed Timelike Loop Achieved}$$