Figure 2

σ x

σ x

(a) Grandfather paradox circuit. If we take 1 to represent “time traveler exists,” and 0 to represent “she doesn’t exist,” then the NOT () 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. 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. Reuse & Permissions