Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency

Closed timelike curves (CTCs) are trajectories in spacetime that effectively travel backwards in time: a test particle following a CTC can interact with its former self in the past. A widely accepted quantum theory of CTCs was proposed by Deutsch. Here we analyze an alternative quantum formulation of CTCs based on teleportation and postselection, and show that it is inequivalent to Deutsch’s. The predictions or retrodictions of our theory can be simulated experimentally: we report the results of an experiment illustrating how in our particular theory the “grandfather paradox” is resolved.

Figure 1

(a) Deutsch’s quantum description of CTCs is based on the consistency condition of Eq. (1), where the unitary $U$ describes an interaction between a chronology-respecting system $A$, initially in the state ${\rho }_A$, and a system $B$ in a CTC. Deutsch demands that the state $\rho$ of $B$ at the input and output of $U$ be equal. Time goes from bottom to top in this and in the following diagrams. (b) P-CTC: postselected quantum teleportation is employed as a description of the closed timelike curve. The bottom curve $\cup$ represents the creation of a maximally entangled state of two systems and the upper curve $\cap$ represents the projection onto the same state.

Figure 2

(a) Grandfather paradox circuit. If we take 1 to represent “time traveler exists,” and 0 to represent “she doesn’t exist,” then the NOT (${\sigma }_x$) operation implies that if she exists, then she “kills her grandfather” and ceases to exist; conversely, if she does not exist, then she fails to kill her grandfather and so she exists. The difference between Deutsch’s CTCs and our P-CTCs is revealed by monitoring the time traveler with two controlled-NOTs (CNOT): the two controlled bits are measured to determine the value of the time-traveling bit before and after the ${\sigma }_x$. Opposite values mean she has killed her grandfather; same values mean she has failed. Using Deutsch’s CTCs, she always succeeds; using P-CTCs she always fails. (b) Unproved theorem paradox circuit. The time traveler obtains a bit of information from the future via the upper CNOT. She then takes it back in time and deposits a copy an earlier time in the same location from which she obtained it (rather, will obtain it), via the lower CNOT. The circuit is unbiased as to the value of the “proof” bit, so it automatically assigns that bit a completely mixed value, as it is maximally entangled with the one emerging from the CTC.

Figure 3

Experiment to illustrate the P-CTC predictions of the grandfather paradox. (a) Quantum circuit. Using a CNOT gate sandwiched between optional Z and X gates, it is possible to prepare all of the maximally entangled Bell states. The Bell state measurement is implemented using a CNOT and a Hadamard. Each of the probe qubits is coupled to the forward qubit via a CNOT gate. (b) Experimental apparatus. The polarization and path degrees of freedom of single photons from a quantum dot are entangled via a calcite polarization-dependent beam displacer (BD1), implementing the CNOT. Half-wave plates (HWP) before and after BD1 implement the optional $Z$ and $X$ gates. To complete the teleportation, the postselection onto $|{\varphi }^{+}⟩$ is carried out by recombining the path degrees of freedom on a polarizing beam splitter (performing a CNOT gate between path and polarization) and then passing the photons through a calcite polarizer set to $45°$ and detecting them. A rotatable HWP acts as a quantum gun, implementing the unitary $U(\theta )=e^{-i\theta {\sigma }_x}$. Removable calcite beam displacers (BD2 and BD3) couple the polarization qubit to two probe qubits encoded in additional spatial degrees of freedom.

Figure 4

Probability that the postselection simulating time travel succeeds and the probes are found in the same state (red circles) or in opposite states (blue diamonds). As the accuracy $\theta$ of the quantum gun increases from 0 to $\pi$, the probability that the teleportation succeeds decreases. Nonetheless, the probability that the probe qubits are found in either the 10 or 01 state (i.e., the kill succeeds) is $0.01\pm 0.04$. Solid curves correspond to theoretical predictions. The theory-experiment discrepancy is due to a $1.1\pm 0.1°$ mismatch between polarizers used for teleportation. The error bars are due to photon counting and background from the cooled CCD. Inset: the teleportation loop is a polarization interferometer with measured visibility $93\pm 3\%$.