Treating time travel quantum mechanically

The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this question by utilizing the quantum circuit formalism has given rise to two theories: Deutschian-CTCs (D-CTCs) and “postselected” CTCs (P-CTCs). In this paper the quantum circuit approach is thoroughly reviewed, and the strengths and shortcomings of D-CTCs and P-CTCs are presented in view of their nonlinearity and time-travel paradoxes. In particular, the “equivalent circuit model”—which aims to make equivalent predictions to D-CTCs, while avoiding some of the difficulties of the original theory—is shown to contain errors. The discussion of D-CTCs and P-CTCs is used to motivate an analysis of the features one might require of a theory of quantum time travel, following which two overlapping classes of alternate theories are identified. One such theory, the theory of “transition probability” CTCs (T-CTCs), is fully developed. The theory of T-CTCs is shown not to have certain undesirable features—such as time-travel paradoxes, the ability to distinguish nonorthogonal states with certainty, and the ability to clone or delete arbitrary pure states—that are present with D-CTCs and P-CTCs. The problems with nonlinear extensions to quantum mechanics are discussed in relation to the interpretation of these theories, and the physical motivations of all three theories are discussed and compared.

Figure 1

Schematic diagram for the standard form circuit described in the text. The $n$ CR and $m$ CV qubits are shown entering the gate labeled by unitary $U$. The double bars represent the time-travel event and may be thought of as two depictions of the same spacelike hypersurface, thus forming a CTC. The dashed lines represent the spacelike boundaries of the region in which time travel takes place; CV qubits are restricted to that region.

Figure 2

Circuit diagram for the unproven theorem circuit of Ref. [16] in the standard form. The book qubit is labeled $B$, mathematician labeled $M$, and time traveler labeled $T$. The input to the circuit is taken to be the pure state ${|0\rangle }_B{|0\rangle }_M$. The notation used for the gates is the standard notation as defined in Ref. [14] and describes a unitary $U=swa{p}_{MT}cno{t}_{BM}cno{t}_{TB}$.

Figure 3

The equivalent circuit model. (a) A circuit equivalent to the standard form of Fig. 1 with $V=swapU$. (b) The same circuit “unwrapped” in the equivalent circuit model. There is an infinite ladder of unwrapped circuits, each with the same input state ${\rho }_i$. To start the ladder, an initial CV state ${\sigma }_0$ is guessed. The CV state of the $N\mathrm{th}$ rung of the ladder is ${\sigma }_N$. The output state ${\rho }_f$ is taken after a number $N\to \infty$ of rungs of the ladder have been iterated.

Figure 4

Schematic diagram illustrating the protocol defining the action of P-CTCs as described in the text. When Alice measures the final state of the $B$ and $A$ qubits to be $|\Phi \rangle$, the standard teleportation protocol would have Bob (who holds the other half of the initially entangled state) then holding the state with which Alice measured the $A$ system. There are two unusual things about this situation: first, that Bob's system is the same one that Alice wants to teleport, just at an earlier time, and, second, that Alice can get the outcome $|\Phi \rangle$ with certainty and so no classical communication to Bob is required to complete the teleportation.