Quantum Time Machine

$$|\psi(t_0)\rangle \;\xrightarrow{\text{unitary evolution}}\; |\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle$$ $$U(t,t_0) = e^{- \frac{i}{\hbar} \hat{H} (t-t_0)}$$ $$\hat{T}|\psi(t)\rangle = |\psi(-t)\rangle$$ $$\rho_{CTC} = \text{Tr}_S\Big[ U (\rho_S \otimes \rho_{CTC}) U^\dagger \Big], \quad \rho_{CTC}^{\rm out} = \rho_{CTC}^{\rm in}$$ $$A_w = \frac{\langle \phi | \hat{A} | \psi \rangle}{\langle \phi | \psi \rangle}$$ $$\boxed{ \text{Time Machine (Quantum Formalism)}: \; |\psi(t)\rangle \sim e^{-i\hat{H}t/\hbar}|\psi(0)\rangle, \; \hat{T}|\psi(t)\rangle = |\psi(-t)\rangle, \; \rho_{CTC} = \text{fixed point} }$$