Figure 2

CTCs and OTCs in presence of ancilla. A represents the system to be sent through the spacetime wormhole, $A\prime$ represents the same system after it passes through the spacetime wormhole, and B some chronology-respecting system initially correlated with A. (a) In general CTCs, temporal self-consistency demands that ρ_CTC satisfies ${\rho }_{\mathrm{CTC}}={\mathrm{Tr}}_{\ne A}[U({\rho }_{AB}^{(\text{in})}\otimes {\rho }_{\mathrm{CTC}})U^{†}]$. (b) In the case of OTCs, this implies that system A has state ${\rho }_{\mathrm{OTC}}={\mathrm{Tr}}_{\ne A}\left[{\rho }_{AB}^{(\text{in})}\otimes {\rho }_{\mathrm{OTC}}\right]={\rho }_A$ after application of the protocol.