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The quantum mechanics of time travel through post-selected teleportation

Seth Lloyd1, Lorenzo Maccone1, Raul Garcia-Patron1, Vittorio Giovannetti2, Yutaka Shikano1,3

"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 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. ... Einstein’s theory of general relativity allows the existence of closed timelike curves, paths through spacetime that, if followed, allow a time traveler – whether human

being or elementary particle – to interact with her former self. ...This paper explores a particular version of closed timelike curves based on combining

quantum teleportation with post-selection. The resulting post-selected closed timelike curves (P-CTCs) provide a self-consistent picture of the quantum mechanics of time-travel. ... Because the theory of P-CTCs rely on post-selection, they provide self-consistent resolutions to such paradoxes: anything that happens in a P-CTC can also happen in conventional quantum mechanics with some probability. Similarly, the

post-selected nature of P-CTCs allows the predictions and retrodictions of the theory to be tested experimentally, even in the absence of an actual general-relativistic closed timelike curve. ... closed timelike curves are a generic feature of highly curved, rotating spacetimes: the Kerr solution for a rotating black hole contains closed timelike curves within the black hole horizon; and massive rapidly rotating cylinders typically are associated with closed timelike curves ... Hawking’s chronology protection postulate, for example, suggests that the conditions needed to create closed timelike curves cannot arise in any physically realizable spacetime [13]. ... Hartle and Politzer pointed out that in the presence of closed timelike curves, the ordinary correspondence between the path-integral formulation of quantum mechanics and the formulation in terms of unitary evolution of states in Hilbert space breaks down ... General relativistic closed timelike curves provide one potential mechanism for time travel, but they need not

provide the only one. Quantum mechanics supports a variety of counter-intuitive phenomena which might allow time travel even in the absence of a closed timelike curve in the geometry of spacetime. ... time travel effectively represents a communication channel from the future to the past. Quantum time travel, then, should be described by a quantum communication channel to the past. A well-known quantum communication channel is

given by quantum teleportation, in which shared entanglement combined with quantum measurement and classical communication allows quantum states to be transported between sender and receiver. We show that if quantum teleportation is combined with post-selection, then the result is a quantum channel to the past. The entanglement occurs between the forward- and backward going parts of the curve, and post-selection replaces the

quantum measurement and obviates the need for classical communication, allowing time travel to take place. The resulting theory allows a description both of the quantum mechanics of general relativistic closed timelike curves, and of Wheeler-like quantum time travel in ordinary spacetime. ... P-CTCs appear to be less pathological [17]. They are based on a different self-consistent condition that states that self-contradictory events do not happen (Novikov principle [29]). Pegg points out that this can arise because of destructive interference of self-contradictory histories

[22] ... any quantum theory which allows the nonlinear process of postselection supports time travel even in the absence of general relativistic

closed timelike curves. ... The mechanism of P-CTCs [17] can be summarized by saying that they behave exactly as if the initial state of

the system in the P-CTC were in a maximal entangled state (entangled with an external purification space) and the final state were post-selected to be in the same entangled state. When the probability amplitude for the transition between these two states is null, we postulate that the related event does not happen (so that the Novikov principle [29] is enforced). ... Note that Deutsch’s formulation assumes that the state exiting the CTC in

the past is completely uncorrelated with the chronology preserving variables at that time: the time-traveler’s ‘memories’ of events in the future are no longer valid.

"If nature somehow provides the nonlinear dynamics afforded by final-state projection, then it is possible for particles (and, in principle, people) to tunnel from the future to the past."

Linear unitarity axioms preclude signal nonlocality, guarantee no cloning and the untappability of quantum entanglement encrypted C^3. This is a Maginot Line in Cyber War. http://www.historylearningsite.co.uk/maginot_line.htm

"it has been pointed out many times before (e.g. see [30, 53]) that when quantum fields inside a CTC interact with external fields, linearity and unitarity is lost."

"any quantum theory that allows the nonlinear process of projection onto some particular state, such as the entangled states of P-CTCs, allows time travel even when no spacetime closed timelike curve exists. ... Non-general relativistic P-CTCs can be implemented by the creation of and projection onto entangled particle-antiparticle pairs. ... projection is a non-linear process that cannot be implemented deterministically in ordinary quantum mechanics, it can 7 easily be implemented in a probabilistic fashion. Consequently, the effect of P-CTCs can be tested simply by performing quantum teleportation experiments, and by post-selecting only the results that correspond to the desired entangled-state output. If it turns out that the linearity of quantum mechanics is only approximate, and that projection onto particular states does in fact occur – for example, at the singularities of black holes [18–21] – then it might be possible to implement time travel even in the absence of a general-relativistic closed timelike curve. The formalism

of P-CTCs shows that such quantum time travel can be thought of as a kind of quantum tunneling backwards in time, which can take place even in the absence of a classical path from future to past."

"It is this demand that closed timelike curves respect both statistics for the time-traveling state together with its correlations with other variables that distinguishes PCTCs from Deutsch’s CTCs. The fact that P-CTCs respect correlations effectively enforces the Novikov principle [29], and, as will be seen below, makes P-CTCs consistent with path-integral approaches to CTCs. ... because of the huge recent interest on CTCs in physics and in computer science (e.g. see [35, 36, 39–43]), it is important to point out that there are reasonable alternatives to the leading theory in the field. We

also point out that our post-selection based description of CTCs seems to be less pathological than Deutsch’s: for example P-CTCs have less computational power and do not require to separately postulate a “maximum entropy rule” [17]. Therefore, they are in some sense preferable, at least from an Occam’s razor perspective. Independent of such questions of aesthetic preference, as we will now show, P-CTCs are consistent with previous path integral formulations of closed timelike curves, whereas Deutsch’s CTCs are not. ... The “conventional” strategy to deal with CTCs using path integrals is to send the system C to a prior time unchanged (i.e. with the same values of x, x? ), while the other system (the chronology-respecting one) evolves normally. ... we briefly comment on the (Aharonov) two-state vector formalism of quantum mechanics [48, 51]. It is based on

post-selection of the final state and on renormalizing the resulting transition amplitudes: it is a time-symmetrical formulation of quantum mechanics in which not only the initial state, but also the final state is specified. As such, it shares many properties with our post-selection based treatment of CTCs. In particular, in both theories it is impossible to assign a definite quantum state at each time: in the two-state formalism the unitary evolution

forward in time from the initial state might give a different mid-time state with respect to the unitary evolution backward in time from the final state. Analogously, in a P-CTC, it is impossible to assign a definite state to the CTC system at any time, given the cyclicity of time there. This is evident, for example, from Eq. (5): in the CTC system no state is assigned, only periodic boundary conditions. Another aspect that the two-state formalism and P-CTCs share is the nonlinear renormalization of the states and probabilities. In both cases this arises because of the post-selection. In addition to the

two-state formalism, our approach can also be related to weak values [48, 52], since we might be performing measurements between when the system emerges from the CTC and when it re-enters it. Considerations analogous to the ones presented above apply. It would be a mistake,

however, to think that the theory of post-selected closed timelike curves in some sense requires or even singles out the weak value theory. Although the two are compatible with each other, the theory of P-CTCs is essentially a ‘free-standing’ theory that does not give preference to one interpretation of quantum mechanics over another. ... a chronology-respecting system in a state that interacts with a CTC using a unitary U will undergo the transformation

where we suppose that the evolution does not happen if

The comparison with (7 ordinary open system QM with dissipation but no CTC) is instructive: there the non-unitarity comes from the inaccessibility

of the environment. Analogously, in (9) the non-unitarity comes from the fact that, after the CTC is closed, for the chronology-respecting system it will be forever inaccessible. The nonlinearity of (9) is more difficult to interpret, but is connected with the periodic boundary conditions in the CTC. ...

V. COMPUTATIONAL POWER OF CTCS

It has been long known that nonlinear quantum mechanics potentially allows the rapid solution of hard problems such as NP-complete problems [56]. The nonlinearities in the quantum mechanics of closed timelike curves is no exception [41–43]. Aaronson and Watrous have shown quantum computation with Deutsch’s closed timelike curves allows the solution of any problem in PSPACE, the set of problems that can be solved using

polynomial space resources [41]. Similarly, Aaronson has shown that quantum computation combined with postselection allows the solution of any problem in the computational class PP, probabilistic polynomial time(where a probabilistic polynomial Turing machine accepts with probability 1

2 if and only if the answer is “yes.”). Quantum computation with post-selection explicitly allows P-CTCs, and P-CTCs in turn allow the performance of

any desired post-selected quantum computation. Accordingly, quantum computation with P-CTCs can solve any problem in PP, including NP-complete problems. Since the class PP is thought to be strictly contained in PSPACE, quantum computation with P-CTCs is apparently strictly less powerful than quantum computation with Deutsch’s CTCs ... Bennett et al. argue, the programmer who is using a Deutschian closed timelike curve as part of her quantum computer typically finds the output of the curve is completely decorrelated from the problem she would like to solve: the curve emits random states.

In contrast, because P-CTCs are formulated explicitly to retain correlations with chronology preserving curves, quantum computation using P-CTCs do not suffer from state-preparation ambiguity. That is not so say that PCTCs are computationally innocuous: their nonlinear nature typically renormalizes the probability of states in an input superposition, yielding to strange and counterintuitive effects. For example, any CTC can be used

to compress any computation to depth one, as shown in Fig. 2. Indeed, it is exactly the ability of nonlinear quantum mechanics to renormalize probabilities from their conventional values that gives rise to the amplification of small components of quantum superpositions that allows the solution of hard problems. Not least of the counter-intuitive effects of P-CTCs is that they could still solve hard computational problems with ease! The ‘excessive’ computational power of P-CTCs is effectively an argument for why the types of nonlinearities that give rise to P-CTCs, if they exist, should only be found under highly exceptional circumstances such as general-relativistic closed timelike curves or black-hole singularities. ... Our hope in elaborating the theory of P-CTCs is that this theory may prove useful in formulating a quantum theory of gravity, by providing new insight on one of the most perplexing consequences of general relativity, i.e., the possibility of time-travel."