Lately, the quant-ph section of the arXiv has been aflurry with papers investigating what would happen to quantum information processing if time travel were possible (see the more recent papers here). I am not sure exactly why this topic has become fashionable, but it may well be an example of the Bennett effect in quantum information research. That is, a research topic can meander along slowly at its own pace for a few years until Charlie Bennett publishes an (often important) paper^{[1]} on the subject and then everyone is suddenly talking and writing about it for a couple of years. In any case, there have been a number of counter-intuitive claims that time travel enables quantum information processing to be souped up. Specifically, it supposedly enables super-hard computational problems that are in complexity classes larger than NP to be solved efficiently^{[2]}^{[3]}^{[4]}^{[5]} and it supposedly allows nonorthogonal quantum states to be perfectly distinguished^{[2]}^{[6]}. These claims are based on two different models for quantum time-travel, one due to David Deutsch^{[7]} and one due to a multitude of independent authors based on post-selected teleportation (this paper^{[8]} does a good job of the history in the introduction).

In this post, I am going to give a basic introduction to the physics of time-travel. In later posts, I will explain the Deutsch and teleportation-based models and evaluate the information processing claims that have been made about them. What is most interesting to me about this whole topic, is that the correct model for time travelling quantum systems, and hence their information processing power, seems to depend sensitively on both the formalism and the interpretation of quantum theory that is adopted^{[9]}. For this reason, it is a useful test-bed for ideas in quantum foundations.

## Basic Concepts of Time-Travel

Everyone is familiar with the sensation of time-travel into the future. We all do it at a rate of one second per second every day of our lives. If you would like to speed up your rate of future time travel, relative to Earth, then all you have to do is take a space trip at a speed close to the speed of light. When you get back, a lot more time will have elapsed on Earth than you will have experienced on your journey. This is the time-dilation effect of special relativity. Therefore, the problem of time-travel into the future is completely solved in theory, although in practice you would need a vast source of energy in order to accelerate yourself fast enough to make the effect significant. It also causes no conceptual problems for physics, since we have a perfectly good framework for quantum theories that are compatible with special relativity, known as quantum field theory.

On the other hand, time travel into the past is a much more tricky and conceptually interesting proposition. For one thing, it seems to entail time-travel paradoxes, such as the grandfather paradox where you go back in time and kill your grandfather before your parents were born, so that you are never born, so that you cannot go back in time and kill your grandfather, so that you are born, so that you can go back in time and kill your grandfather etc. (see this article for a philosophical and physics-based discussion of time travel paradoxes). For this reason, many physicists are highly sceptical of the idea that time travel into the past is possible. However, General Relativity (GR) provides a reason to temper our skepticism.

## Closed Timelike Curves in GR

It has been well-known for a long time that GR admits solutions that include closed timelike curves (CTCs), i.e. world-lines that return to their starting point and loop around. If you happened to be travelling along a CTC then you would eventually end up in the past of where you started from. Actually, it is a bit more complicated than that because the usual notions of past and future do not really make sense on a CTC. However, imagine what it would look like to an observer in a part of the universe that respects causality in the usual sense. First of all, she would see you appear out of nowhere, claiming to have knowledge of events that she regards as being in the future. Some time later she would see you disappear out of existence. From her perspective it certainly looks like time-travel into the past. What things would feel like from your point of view is more of a mystery, as the notion of a CTC makes a mockery of our usual notion of “now”, i.e. it is a fundamentally block-universe construct.

The possibility of CTCs in GR was first noticed by Willem van Stockum in 1937^{[10]} and later by Kurt Gödel in 1949^{[11]}. Perhaps the most important solution that incorporates CTCs is the Kerr vacuum, which is the solution that describes an uncharged rotating black hole. Since most black holes in the universe are likely to be rotating, there is a sense in which one can say that CTCs are generic. The caveat is that the CTCs in the Kerr vacuum only occur in the interior of the black hole so that the physics outside the event horizon respects causality in the usual sense. Many physicists believe that the CTCs in the Kerr vacuum are mathematical artifacts, which will perhaps not occur in a full theory of quantum gravity. Nevertheless, the conceptual possibility of CTCs in General Relativity is a good reason to look at their physics more closely.

There have been a few attempts to look for solutions of GR that incorporate CTCs that a human being would actually be able to travel along without getting torn to pieces. This is a bit beyond my current knowledge, but, as far as I am aware, all such solutions involve large quantities of negative energy, so they are unlikely to exist in nature and it is unlikely that we can construct them artificially. For this reason, CTCs are currently more of a curiosity for foundationally inclined physicists like myself than they are a practical method of time-travel.

## Other Retrocausal Effects in Physics

Apart from GR, other forms of backwards-in-time, or retrocausal, effect have been proposed in physics from time to time. For example, there is the Wheeler-Feynman absorber theory of electrodynamics, which postulates a backwards-in-time propagating field in addition to the usual forwards-in-time propagating field, and Feynman also postulated that positrons might be electrons travelling backwards in time. There is also Cramer’s transactional interpretation of quantum theory^{[12]}, which does a similar thing with quantum wavefunctions, and the distinct, but conceptually similar, two-state vector formalism of Aharonov and collaborators^{[13]}. Finally, retrocausal influences have been suggested as a mechanism to reproduce the violations of Bell-inequalities in quantum theory without the need for Lorentz-invariance violating nonlocal influences^{[14]}.

However, none of these proposals are as compelling an argument for taking the physics of time-travel into the past seriously as the existence of CTCs in General Relativity. This is because, none of these theories gives provides a method for exploiting the retrocausal effect to actually travel back in time. Also, in each case, there is an alternative approach to the same phenomena that does not involve retrocausal influences. Nevertheless, it is possible that the models to be discussed have applications to these alternative approaches to physics.

## Consistency Constraints and The Interpretation of Quantum Theory

Any viable theory of time travel into the past has to rule out things like the grandfather paradox. Consistency conditions have to be imposed on any physical model to so that time-travel cannot be used to alter the past. This raises interesting questions about free will, e.g. what exactly stops someone from freely deciding to pull the trigger on their grandfather? Whilst these questions are philosophically interesting, physicists are more inclined to just lay out the mathematics of consistency and see what it leads to. The different models of quantum time travel are essentially just different methods of imposing this sort of consistency constraint on quantum systems.

That is pretty much it for the basic introduction, but I want to leave you with a quick thought experiment to illustrate the sort of quantum foundational issues that come up when considering time-travel into the past. Suppose you prepare a spin-\(\frac{1}{2}\) particle in a spin up state in the z direction and then measure it in the x direction, so that it has a 50-50 chance of giving the spin up or spin down outcome. After observing the outcome you jump onto a CTC, travel back into the past and watch yourself perform the experiment again. The question is, would you see the experiment have the same outcome the second time around?

A consistency condition for time travel has to say something like “the full ontic state (state of things that exist in reality) of the universe must be the same the second time round as it was the first time round”, albeit that your subjective position within it has changed. If you believe, as many-worlds supporters do, that the quantum wavefunction is the complete description of reality then it, and only it, must be the same the second time around. Therefore, it must be the case that the probabilities are still 50-50 and you could see either outcome. This is not inconsistent because the many-worlds supporters believe that both outcomes happened the first time round in any case. If you are a Bohmian then the ontic state includes the positions of all particles in addition to the wavefunction and these, taken together, can be used to determine the outcome of the experiment uniquely. Therefore, a Bohmian must believe that the measurement outcome has to be the same the second time around. Finally, if you are some sort of anti-realist neo-Copenhagen type then it is not clear exactly what you believe, but, then again, it is not clear exaclty what you believe even when there is no time-travel.

There are some subtleties in these arguments. For example, it is not clear what happens to the correlations between you and the observed system when you go around the causal loop. If they still exist then this may restrict the ability of the earlier version of you to prepare a pure state. On the other hand, perhaps they get wiped out or perhaps your memory of the outcome gets wiped. The different models for the quantum physics of CTCs differ on how they handle this sort of issue, and this is what I will be looking at in future posts. If you have travelled along a CTC and happen to have brought a copy of these future posts with you then I would be very grateful if you could email them to me because that would be much easier for me than actually writing them.

‘Till next time!

## References

- Bennett, C. H. et. al. (2009). “Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems”. Phys. Rev. Lett. 103:170502. eprint arXiv:0908.3023. [↩]
- Brun, T. A. and Wilde, Mark M. (2010). “Perfect state distinguishability and computational speedups with postselected closed timelike curves”. eprint arXiv:1008.0433. [↩] [↩]
- Aaronson, S. and Watrous, J. (2009). Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A 465:631-647. eprint arXiv:0808.2669. [↩]
- Bacon, D. (2004). Quantum Computational Complexity in the Presence of Closed Timelike Curves. Phys. Rev. A 70:032309. eprint arXiv:quant-ph/0309189. [↩]
- Brun, T. A. (2003). Computers with closed timelike curves can solve hard problems. Found. Phys. Lett. 16:245-253. eprint arXiv:gr-qc/0209061. [↩]
- Brun, Todd A., Harrington, J. and Wilde, M. M. (2009). “Localized closed timelike curves can perfectly distinguish quantum states”. Phys. Rev. Lett. 102:210402. eprint arXiv:0811.1209. [↩]
- Deutsch, D. (1991). “Quantum mechanics near closed timelike lines”. Phys. Rev. D 44:3197—3217. [↩]
- Lloyd, S. et. al. (2010). “The quantum mechanics of time travel through post-selected teleportation”. eprint arXiv:1007.2615 [↩]
- I should mention that Joseph Fitzsimons (@jfitzsimons) disagreed with this statement in our Twitter conversations on this subject, and no doubt many physicists would too, but I hope to convince you that it is correct by the end of this series of posts. [↩]
- Stockum, W. J. van (1937). “The gravitational field of a distribution of particles rotating around an axis of symmetry”. Proc. Roy. Soc. Edinburgh A 57: 135. [↩]
- Kurt Gödel (1949). “An Example of a New Type of Cosmological Solution of Einstein’s Field Equations of Gravitation”. Rev. Mod. Phys. 21: 447. [↩]
- Cramer, J. G. (1986). “The transactional interpretation of quantum mechanics”. Rev. Mod. Phys. 58:647-687. [↩]
- Aharonov, Y. and Vaidman, L. (2001). “The Two-State Vector Formalism of Quantum Mechanics: An Updated Review”. in “Time in Quantum Mechanics”, Muga, J. G., Sala Mayato, R. and Egusquiza, I. L. eprint arXiv:quant-ph/0105101. [↩]
- For example, see Price, H. (1997). “Time’s Arrow and Archimedes’ Point”. OUP. [↩]

Looking forward to the rest of the series!

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As far as I can understand them close timelike curves are completely impractical as a means of time travel since whatever enters such curve can by definition never leave it – otherwise the curve would not be closed.

It might work as a means for transferring information into the past.

I think the easiest way to do away with paradoxes is to invoke temporal interference – wavefunctions interfering in time so that for example detecting this particle here and now mean that particle cannot be there later or something like that. I admit it’s a bit hard to imagine how it would save grandpa but it should.

Your statement about CTCs not allowing time-travel does not follow just from the definition. If the CTC is a geodesic and you are just imagining a small test particle free-falling along it then of course it can never leave the loop. On the other hand, if you have a source of energy, e.g. a rocket ship, then you can accelerate for a period of time and modify which geodesic you are travelling along subsequently. Therefore, what you say is correct only if every world-line that intersects a space-time event that is on a CTC is itself closed. Although I am not an expert on General Relativity, I am pretty sure this isn’t true because I have heard about models of wormholes that allow for time travel and if you travel through one then you can make a closed loop by travelling to the space-time event that you started from, but I do not see why you would be compelled to re-enter the same wormhole the second time around.

A closed timelike curve is a worldline that is closed and returns to its starting point, so if it were your wordline you would by definition have to circle the closed part forever at least assuming the static 4D block universe of GR, since once you made the first circle you would then only repeat what already happened to you – you could not change it.

What I think you mean is following only part of the CTC trajectory and leaving before you make the first full loop, then I agree you could leave but since your wordline would not overlap anywhere it wouldn’t technically be a closed timelike curve. This is what I originally meant, although I guess you could still say that you have exploited the presence of CTCs in spacetime to make your time travel.

Paul, I think you misunderstand what the CTC refers to. The existence of closed timelike curves are a property of the spacetime, not of a the trajectory of a particle.

While “the existence of CTCs” are a property of spacetime, a CTC is defined as a closed wordline so it is a special kind of a “trajectory of a particle” – one which returns to the exact point/event in spacetime where it started.

Actually this very distinction is the whole point of my second post above, while you can say that the existence of CTCs implies that time travel is possible, a practical time travel cannot employ CTCs themselves because the traveller would end up trapped in it forever.

Paul, I don’t quite think that is true. What is often meant is that there exists a closed time-like curve on the space time, and the consequences of this for information processing, but in reality it makes little difference. In Deutsch’s model, there is indeed a qubit residing on a closed time like curve (let’s call this C), which interacts with other qubits. Note, however, that this does potentially allow for time-travel: We take two ancilla qubits, A and B, at time T1 in the lab frame C and A are swapped. At T2>T1 B and C are swapped. But when the consistency condition is applied we find that qubit A must immediately obtain the state of C when the first swap is applied at T1, but that C retains the same state until T2. This pretty clearly fits any reasonable definition of time-travel, and the information does not get “stuck” on the CTC.

“This pretty clearly fits any reasonable definition of time-travel, and the information does not get “stuck” on the CTC.”

To me travel and communication are two different things, I did make this distinction in the first post and I did say that CTCs “might work as a means for transferring information into the past.” I only object to CTCs being good for practical travel which involves material objects like a time traveler for example.