Thursday, October 13, 2005

Quantum Time Machines: What? Why? and (maybe) How?

Tim Ralph is in Tokyo this week, and is giving a talk today. Here's
the abstract:


Speaker: Prof.Tim C. Ralph
Centre for Quantum Computer Technology,
University of Queensland,
Australia.


Title: Quantum Time Machines: What? Why? and (maybe) How?

Abstract:

Whether time travel into the past is possible is an undecided physical
question [1]. Recently it has been noted that certain models of time
travel for quantum particles do not lead to the same difficult
paradoxes that arise for classical particles [2]. Furthermore the
types of quantum evolutions predicted for these "quantum time
machines" could give rise to a "super" quantum computer, able to solve
problems thought to be intractable by any other means [3]. In this
talk I will discuss time machines in general, how quantum mechanics
avoids the paradoxes and the unusual evolutions predicted. I will
then argue that the requirements for realizing such machines are not
as stringent as previously thought and I will propose "horizon
technology" experiments which could test these ideas [4].


[1] M.S.Morris, K.P.Thorne and U.Yurtsever, Phys.Rev.Lett.61, 1446 (1988).
[2] D.Deutsch, Phys.Rev.D 44, 3197 (1991).
[3] D.Bacon, Phys.Rev.A, 70, 032309 (2004).
[4] T.C.Ralph, quant-ph/0510038.



With an abstract like that, you'd have to beat with me a stick to keep me away from the talk. Now, I only know a little about relativity - what every Caltech freshman knows (special relativity) and any fan of popular science writing knows (e.g., Kip Thorne's fabulous Black Holes and Time Warps). I know squat about the real math of general relativity, but I'm familiar with the basic implications of the curvature of space, black holes, wormholes, etc. So, should be a fun but challenging talk.

To prepare, I spent a little time this morning reading Tim's paper (quant-ph/0510038) and took a quick look at Dave Bacon's recent paper. Dave's paper is apparently a pretty major one. Building on work by Deutsch, Brun, and others, he shows that the existence of closed timelike curves (CTCs) would allow NP-complete problems to be solved on a quantum computer using only polynomial resources.

What the heck is a closed timelike curve? It's a path through a wormhole that comes back to where you started. Special relativity treats space and time as a four-dimensional space called spacetime, and establishes rules for paths in that space. You can never reach another point in spacetime that would require you to travel faster than light to get there. General relativity, which introduces the curvature of space, in its extreme appears to allow the creation of wormholes - places in space that, if you step through, take you to another distant point in space. Travelling through a wormhole can allow you to travel into your own past, coming back to where you started, and creating a "closed timelike curve". So Dave showed that travelling through a wormhole will allow you to solve NP-complete problems efficiently on a quantum computer. (We won't talk about grandfather paradoxes and the like, but apparently quantum mechanics helps there.)

Moving on, Tim's paper is lucid and remarkably easy to read for a non-specialist like me. He shows that you can take an entangled pair of quanta (an EPR pair), relativistically accelerate one and bring it back to the starting point, and like the pedagogical twins, one particle will now be "older" than the other. Using the standard quantum techniques of teleportation, he then creates a state that's something a little like a wormhole, essentially a loop in time. You can't use this to violate causality or learn about the future, but it does allow nonlinear evolution of the quantum system. Dave's system depends on the nonlinearity created by the wormhole; here Tim has done it without one. Fascinating.

Okay, time to pack up and head to the seminar and see whether I've correctly interpreted Tim, and what other features of this there are that I haven't thought of...

(From this point, it's just the notes I typed during the seminar, so apologies if they seem a little abrupt or semi-coherent.)

Morris, Thorne, Yurtsever, PRL 61

Where are all the time tourists? There's a plaque in Perth that says, "In the event that the transportation of life from the future" becomes possible, please show up at March 31, 2005, in Perth. No one did. (Reminds me of the MIT conference.) Requiring that a time machine receiver be built first eliminates the problem of the lack of visible travellers, but still doesn't eliminate the standard causality problems.

Tim actually likes Sagan's Contact. Wormholes were developed by Einstein and Rosen, so they are also called Einstein-Rosen bridges, but in their solution the wormhole collapses too quickly to be used. Kip Thorne showed that if you thread a wormhole with exotic matter, it might stay open long enough to let regular matter pass through. Thorne showed that many of the possible paradoxes resolve themselves, but whether they all resolve comfortably is unknown.

You get a nice little paradox if you use classical bits and a CNOT gate. Start with a one on the target bit, run it through the CNOT gate, and take that output rho, shove it through your time machine into the past, then use that as the control line for the CNOT gate. There's no possible consistent result. (Think about the states where rho comes out of the time machine as a one or a zero.)

However, if you do this with qubits, there is a solution. rho = 1/2(|0><0| + |1><1|) works. Deutsch then generalized and showed that any unitary evolution U has a solution without creating classical paradoxes, provided that you keep your particle that goes through the wormhole in a box and neither measure nor prepare it during the time travel interval.

The wormhole introduces a nonlinearity. Non-relativistic ("classical") quantum mechanics, of course, is linear. So is there a way to use that nonlinearity now added? Yes, in certain regions, it creates a faster growth of the trace distance between two states. This is where the power comes from that Bacon used to solve NP-complete problems.

Of course, we don't have a source of wormholes to experiment with, and don't even know how to build them. So what about a stochastic time machine? One that lights up one of four possible lights (labeled I, X, Z, and XZ) every time you shove a particle into to send it into the past. If you're familiar with quantum teleportation, you'll immediately recognize those as the operators you need to apply to retrieve the teleported state, once you've measured the qubit at the source.

During the time travel interval (between the past time where you receive the particle, and the future time where you sent it), what you have is a mixed state. Now there are no paradoxes, even if you observe the particle. And you can compute with it. But your real, true final answer isn't known until after the (local) time that you sent the particle into the time machine.

Teleportation can be considered a "stochastic wormhole". Do the same thing you would with a real wormhole - accelerate one of the particles at relativistic speeds for a while, then bring it back where you started - and you have a stochastic time machine.

What does it all mean? It seems that quantum systems are better behaved with respect to time travel than classical machines. A stochastic quantum time machine still leads to interesting and potentially useful quantum evolutions whilst avoiding ad hoc assumptions. Teleportation using time displaced entanglement appears equivalent to a stochastic time machine. These experiments are hard, but not too hard.

Question:What does increasing the overlap do to security proofs of QKD and things of that nature? Answer:Don't know. Would be interesting to check...

Question:Increasing the overlap, doesn't that have implications for cloning and superluminal communication? Answer:You essentially "lose contact" with other qubits (you have to trace out over other qubits, because the operation is on the reduced density operator). So, no, you don't get superluminal. (See Deutsch.)

It startles and intrigues me no end that quantum mechanics, information theory, and general relativity converge in such interesting ways.

See this auction!

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