Quantum mechanics of time travel through post-selected teleportation

This paper discusses the quantum mechanics of closed-timelike curves (CTCs) and of other potential methods for time travel. We analyze a specific proposal for such quantum time travel, the quantum description of CTCs based on post-selected teleportation (P-CTCs). We compare the theory of P-CTCs to previously proposed quantum theories of time travel: the theory is inequivalent to Deutsch’s theory of CTCs, but it is consistent with path-integral approaches (which are the best suited for analyzing quantum-field theory in curved space-time). We derive the dynamical equations that a chronology-respecting system interacting with a CTC will experience. We discuss the possibility of time travel in the absence of general-relativistic closed-timelike curves, and investigate the implications of P-CTCs for enhancing the power of computation.

Figure 1

Description of closed-timelike curves through teleportation. a) Conventional teleportation: Alice and Bob start from a maximally entangled state shared among them represented by “$\bigcup _{}^{}$”. Alice performs a Bell measurement M on her half of the shared state and on the unknown state $|\psi ⟩$ she wants to transmit. This measurement tells her which entangled state the two systems are in. She then communicates (dotted line) the measurement result to Bob who performs a unitary V on his half of the entangled state, obtaining the initial unknown state $|\psi ⟩$. The numbers refer to the systems as indicated by the subscripts in Eq. (3). b) Post-selected teleportation: the system in state $|\psi ⟩$ and half of the Bell state “$\bigcup _{}^{}$” are projected onto the same Bell state “$\bigcap _{}^{}$.” This means that the other half of the Bell state is projected into the initial state of the system $|\psi ⟩$ even before this state is available.

Figure 2

Closed-timelike loops can collapse the time depth of any circuit to one, allowing to compute any problem not merely efficiently, but instantaneously.