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

"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. ...

As in

all versions of time travel, closed timelike curves embody

apparent paradoxes, such as the grandfather paradox, in

which the time traveller inadvertently or on purpose performs

an action that causes her future self not to exist.

Einstein (a good friend of G¨odel) was himself seriously

disturbed by the discovery of CTCs [11]. 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. ...

The G¨odel universe consists of a cloud of

swirling dust, of sufficient gravitational power to support

closed timelike curves. Later, it was realized that 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 [2, 8, 12]. The

topic of closed timelike curves in general relativity continues

to inspire debate: Hawking’s chronology protection

postulate, for example, suggests that the conditions

needed to create closed timelike curves cannot arise in

any physically realizable spacetime ...

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 [5, 7]. Morris

et al. explored the quantum prescriptions needed to construct

closed timelike curves in the presence of wormholes,

bits of spacetime geometry that, like the handle

of a coffee cup, ‘break off’ from the main body of

the universe and rejoin it in the the past [4]. Meanwhile,

Deutsch formulated a theory of closed timelike

curves in the context of Hilbert space, by postulating

self-consistency conditions for the states that enter and

exit the closed timelike curve ...

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. ...

We start from the prescription that time travel

effectively represents a communication channel from the

future 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. ...

entanglement and projection can give rise to closed timelike

curves ...

Deutsch’s theory has recently

been critiqued by several authors as exhibiting

self-contradictory features [33–36]. By contrast, although

any quantum theory of time travel quantum mechanics

is likely to yield strange and counter-intuitive results,

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]. ...

in addition to

general-relativistic CTCs, our proposed theory can also

be seen as a theoretical elaboration of Wheeler’s assertion

to Feynman that ‘an electron is a positron moving

backward in time’ [16]. In particular, 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 chronologypreserving

variables at that time: the time-traveler’s

‘memories’ of events in the future are no longer valid.

The primary conceptual difference between Deutsch’s

CTCs and P-CTCs lies in the self-consistency condition

imposed. ...

It seems that, based on what is currently known on

these two approaches, we cannot conclusively choose PCTCs

over Deutsch’s, or vice versa. Both arise from

reasonable physical assumptions and both are consistent

with different approaches to reconciling quantum mechanics

with closed timelike curves in general relativity.

A final decision on which of the two is “actually the case”

may have to be postponed to when a full quantum theory

of gravity is derived (which would allow to calculate

from first principles what happens in a CTC) or when

a CTC is discovered that can be tested experimentally. ...

[Aharonov's theory]

Here we briefly comment on the 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 ...

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. ...

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. ...

when quantum fields inside a CTC interact with

external fields, linearity and unitarity is lost. ...

Hartle notes

that CTCs might necessitate abandoning not only unitarity

and linearity, but even the familiar Hilbert space

formulation of quantum mechanics [7]. Indeed, the fact

that the state of a system at a given time can be written

as the tensor product states of subsystems relies crucially

on the fact that operators corresponding to spacelike

separated regions of spacetime commute with each

other. When CTCs are introduced, the notion of ‘spacelike’

separation becomes muddied. The formulation of

closed timelike curves in terms of P-CTCs shows, however,

that the Hilbert space structure of quantum mechanics

can be retained. ...

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. ...

projection is a non-linear process that cannot be implemented deterministically

in ordinary quantum mechanics, it can 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. ...

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 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 ...

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. ...

We have extensively argued that P-CTCs are physically

inequivalent to Deutsch’s CTCs. In Sec. II we

showed that P-CTCs are compatible with the pathintegral

formulation of quantum mechanics. This formulation

is at the basis of most of the previous analysis

of quantum descriptions of closed time-like curves, since

it is particularly suited to calculations of quantum mechanics

in curved space time. P-CTCs are reminiscent of,

and consistent with, the two-state-vector and weak-value

formulation of quantum mechanics. It is important to

note, however, that P-CTCs do not in any sense require

such a formulation. ...

we have argued that, as Wheeler’s picture

of positrons as electrons moving backwards in time suggests,

P-CTCs might also allow time travel in spacetimes

without general-relativistic closed timelike curves. 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.

Finally, in Sec. V we have seen that P-CTCs are computationally

very powerful, though less powerful than the

Aaronson-Watrous theory of Deutsch’s CTCs.

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."