Gilles Brassard has a very nice article in the Commentary section of Nature, “Is information the key?” From the abstract:
Quantum information science has brought us novel means of calculation and communication. But could its theorems hold the key to understanding the quantum world at its most profound level? Do the truly fundamental laws of nature concern — not waves and particles — but information?
The article is well worth reading.
The basic question asked in the paper is whether or not it is possible to derive quantum theory from basic rules about information plus a little bit more. Thus for instance, one can ask, whether it is possible to derive quantum theory by assuming things like no superluminal communication plus no bit commitment (two properties of quantum theory as we understand it today.) To date, there have been some very nice attempts to move this task forward. In particular Brassard mentions the work of Bub, Clifton and Halvorson which is very nice. However, my beef with all such derviations I’ve seen so far is that their assumptions are too strong. For example in the Bub et al work, they assume theory must be described within a C*-algebraic framework. And this assumption just hides too much for me: such assumptions basically are assumption of the linearity of the theory and don’t really shed light on why quantum theory should act in this manner. Linearity, for me, is basically the question “why amplitudes and not probabilities?” This, I find, is a central quantum mystery (well not a mystery, but something I’d like to see a reason given for in the same way that if I had been around in 1900, I would have wanted to see an explanation for the Lorentz transform, which is what Einstein, so beautifully, did.) On the other hand, the fact that one can make these assumptions and derive quantum theory or quantum-like behavior is extremely suggestive and I would be happy if lots of people started thinking about this question.
Personally, I’ve already gone through a stage where I thought this might be the approach to undestanding quantum theory and moved to another stage (just as I have, throughout my life, loved every single interprtation. I even have a strong memory of sitting in a car on a vacation with my family, probably when I was in high school, where I distinctly remember understanding the many-worlds interpretation of quantum theory! On the other hand, I also have a vivid memory of waking up one night while I was an undergrad at Caltech and having an absolutely impossible to disprove reason for why there is evil in the world. Strange feelings, those. “The moment of clarify faded like charity does.”) In work I did with Ben Toner, we showed a protocol for simulating the correlations produced by projective measurements on a singlet using shared randomness and a single bit of communication. For a long time, Ben and I wondered whether we could derive this protocol via more basic assumptions. For example, is there a simple game for which the protocol with one bit of communication is the best strategy (this game also being best solved by bare, unaided with communication, quantum theory?) Of course, one can always define a game such that these correlations and the protocol are the best, but that is cheating: we wanted a simple game to go along with our simple protocol. Alas we could never find such a game or a more basic set of principles from which to derive our protocol. As a funny note, we called the game we were searching for “Warcraft.” And when we were looking for a game which the optimal strategy would yeild all of quantum theory we called it simply “The Game.” What is “The Game” at which quantum theory is the optimal strategy?
After working on “The Game” and related ideas, I’ve become less convinced that this is the proper approach to take. Why? Well mostly due to the structure of the protocol we came up with for simulating the singlet quantum correlations. The important observation, I now believe, about this protocol is its geometric nature. If you look at the protocol (quant-ph/0304076) what is interesting about it, to me, is the beautiful geometry of the protocol. I should mention that recently an reformulation of our protocol has appeared quant-ph/0507120 by Degorre, Laplante, and Roland which is very nice and also demonstrates how simple the geometry involved in the protocol is. So why do I focus on this geometric aspect? Well because I think that the ultimate explanation for why quantum theory is the way it is must, in some way provide answers to the question of hidden variables in quantum theory (prejudice number 1) and that the only way in which I know to get around such obstacles is to muck with the topology of spacetime (prejudice number 2), and thus an attempt to understand our protocol in terms of changes in the topology of spacetime is the proper route to take. However, I haven’t yet succeeded in recasting our protocol in these terms. Perhaps some crazy wonderkid out there can easily see how to do this! (Alternatively I wouldn’t be surprised if my prejudice number two is somehow replaced with particularly strange reasoning about the nature of time. Here I am thinking somewhat of the transactional interpretations of quantum theory.)
Anyway, understanding why quantum theory is the way it is, is either one of the greatest mysteries of physics, or a dead end that will never lead to any experiments. I hope for the former, but, can’t quite convince myself that it can’t be the latter.
I said something similar before in response to these sentiments; maybe I can rephrase it differently. I am not at all convinced that quantum probability is any genuine mystery, and I see no reason to hope one way or the other. It is what it is.
But it is true that before before I understood quantum information theory (and before quantum computation existed and before I believed that too), I was left with the feeling that there must be some interesting questions in this area. Well, now we know what they are. The interesting questions are things like BQP vs BPP, the Holevo conjecture, can we build quantum computers, and so on. I consider these real questions to be an effective and sorely needed antidote to non-questions like what is the right “interpretation” of quantum theory.
Or, at best, interpretation is a question of exposition and intuition, and therefore a human question rather than a physics question. I personally accept quantum probability as operator algebraists do: as the straightforward non-commutative generalization of classical probability. Maybe my hopes are not quite completely neutral. Rather, I have some hope that some such characterization can terminate wonderment. (Or perhaps relegate it to string theorists.)
But there aren’t places where we are unsure of what quantum theory says. It isn’t like QED, where people know (almost to the point of mathematical rigor) that it becomes inconsistent in certain limits. Quantum probability doesn’t curl at the edges like this. It is always possible that quantum probability will be extended to something more truthful, since after all this is what happened to classical probability. But you are really speculating about the unforseeable here.
Quantizing gravity can be given a more flexible meaning than the one use assign it here. It can mean any reconciliation whatsoever of quantum probability and non-quantum general relativity. This is a meaning of “quantum gravity” that I think is what you want.
It just so happens that no one has had any good ideas for changing the quantum side in trying to mesh “quantum” and “gravity”. There is one really interesting idea for changing the gravity side, namely string theory. Do you think that this is too one-sided? Here I really am neutral. Changing only the quantum side and not the gravity side seems hopeless. I’m not sure why I should be happy or unhappy if the eventual theory of everything is a modification of only the gravity side and not both sides.
Anyway it’s a question for quantum gravity theorists and not quantum information theorists.
“And when we were looking for a game which the optimal strategy would yeild all of quantum theory we called it simply “The Game.†What is “The Game†at which quantum theory is the optimal strategy?”
Could it be the Glass Bead Game?
Forgive my ignorance in the next paragraph:
I know that the opposite (which is probably much easier) has been shown to some extent (Seth Lloyd?) but does the non-existence of superluminal transfer lead to the linearity of QM? This of course reminds me of the (NP != P)=>QM debate but man we are supposedly phsyicists (or chemists or ee people)… In a way isn’t this the anti-thesis of quantum gravity?
By the way, too much Hesse has been shown to have adverse effects on human health.
“I know that the opposite (which is probably much easier) has been shown to some extent (Seth Lloyd?) but does the non-existence of superluminal transfer lead to the linearity of QM?”
One has to be a little bit careful with this question. Superluminal transfer is actually possible in nonrelativistic QM but not by local operations on subsystems that are combined via the tensor product. The latter is often what people are referring to when they talk about superluminal transfer and indeed it does imply linearity (see quant-ph/0102125).
http://arxiv.org/abs/quant-ph/0204106 is a good paper by Adrian Kent where he shows that it is false that the non-existence of superluminal transfer lead to the linearity of QM. He does this by constructing a maximal state readout for each point in space that isn’t inconsistent with preserving locality. He then allows (classical) operations based upon this local state readout. This gives nonlinear evolutions, but without any collapse or reduction.
“Or, at best, interpretation is a question of exposition and intuition, and therefore a human question rather than a physics question.”
Greg, this is your interpretation of what foundations of quantum theory means, and I think most people who work on foundations are thinking along these lines (i.e. a human question), but the kind of foundations I am talking about is much different.
I take foundations to also include understand how and where quantum theory might break down, and what this might tell us about any deeper theory. For the same reasons that we study string theory (not testable now, but if we understand it and it is correct and can turn it into a testable theory), I think we can justify studying the foundations of quantum theory. In prarticular, as I’ve said before, understanding the relationship between quantum theory and general relativity is something I think well worth thinking about from a foundations point of view. This doesn’t mean quantizing gravity, but instead means understanding how some joint theory of can give rise to both limits where each theory works excellently. I don’t buy most of the arguments about why we _must_ quantize gravity: they all, to me, have logical weak points.
So I think we agree: if foundations means simply reintreting quantum theory, then I’m with you…spend your time working on finding efficient algorithms for the hidden subgroup problem. But if foundations also means understanding the limits where we are unsure of what quantum theory says, then I think this is interesting physics. Certainly in my above essay I translate this latter question over into language that sounds more like the former. But really I’m more interested in the latter, but am not sure that one cannot get to the latter without also thinking deeply about the former. Oh boy, what did I just write?
It is interesting that you bring up QED. Because we generally think about laying electrodynamics on top of quantum theory, we generally attribute the breakdown (at small length scales) of QED to the physics. But couldn’t this equally well be due to a breakdown in quantum theory? I’ve never understood why the breakdown MUST come from the physics side. Perhaps there is a good reason…I’d love to hear it!
“It just so happens that no one has had any good ideas for changing the quantum side in trying to mesh “quantum†and “gravityâ€.” Perhaps I overstated here: it’s also possible that you don’t need to change the quantum side, but that you can get the gravity side (or an extension thereof) to give you the quantum side.
But I also agree that being a quantum information theorist gives me little or no right to blab about this stuff! Except there are times when I hear certain high energy physicists talk about quantum theory and I have no idea what beast they are describing. Certainly not our beautiful quantum lady of quantum computing 😉
One statement in the Broussard article confuses me. He says, (or rather, quotes Deutsch as saying), “After we reported the first experimental realization of quantum cryptography, Deutsch wrote in New Scientist: “Alan Turing’s theoretical model is the basis of all computers. Now, for the first time, its capabilities have been exceeded.”
This coupled with his earlier statement, “are there information-processing tasks that are impossible even in principle in the classical world, but that become possible through quantum mechanics?”
suggests that quantum crypto indeed breaks not only the effective Church-Turing thesis, but the C-T thesis itself, by presenting a task that is “impossible” in a classical setting, but can be performed with a quantum computer.
I am fairly sure that this is not what is implied, and that I am confused, but I was wondering if someone could clarify this ?
Dave,
There is an article by Joe Polchinski in PRL commenting on Weinberg’s models of non-linear QM, showing that in such models you can build an “Everett phone” — i.e., communicate between branches of the many-worlds wavefunction. (This was many years ago — around 1990?)
Weinberg derives strong experimental limits (from atomic physics) on non-linearity, but Polchinski’s observation is more fundamental and is perhaps what you were looking for to justify exact linearity.
Ah, I see. this is a rather confusing way of putting things though. Where I come from, the fact that you can “simulate” a quantum algorithm in exponential time appears to end the debate over C-T, but then again, I am not a quantum expert.
> Until “gravity information theorists†exist
There is something similar already, called Information Geometry; Proposals on how to get to a quantum theory of gravitation have been made also.
Suresh: The problem is in what these physicists call “classical.” When they say that something cannot be done in the classical world, they are saying that in a similar setup where we use only classical systems we cannot do this. But that doesn’t break the Church-Turing thesis. Why? Because it is still possible to simulate the quantum experiments with a different physical setup. So it is totally consistent to have quantum systems doing things which no classical system can do for a particular experimental setup, and yet, if we really care only about the computability of the process then a different classical system can simulate the quantum system. So, no the CT thesis does not appear to be broken (note there are claims from a few that it does get broken, but I have a few issues with these claims. Ack, I’d better keep my mouth shut.)
Great comments! I was going to point out the Kent paper, but also I also have strong opinions about trying to show an inconsistency arrises when you modify quantum theory. Why? Well because if such modifications arise, then they will arise in setting where we haven’t performed experiments. And what one normally does is adjust quantum theory in some way (say by making it nonlinear) and then combine this with the unmodified version to show some impossibility (like superluminal communication) arises. But in this argument one needs to justify why you can pull the nonlinearity down to the standard theory: it seems possible (necessary?) to have a theory in which exactly that sort of task simply fails.
The problem is that we live too closely to quantum theory. Thinking about how to modify it inevitably leads us to modifications which are still expressible within the framework of quantum theory. I suspect, for reasons like the Polchinski paper, that such modifications are ultimately doomed (although I am not certain about this.)
BTW, this is all good an fun, and what is cool is that it is possible to obtain constructive results (like Polchinski’s), but I’d better get back to my quantum computing stuff before all you smart people solve all the low lying fruit 😉
“Is Geometry the Key”?
I do not know (after all I’m a farmer, and not a theorist). But there is an interesting quote, taken from a paper by Olivier Costa de Beauregard. ‘Algebraic nonseparability entails geometric nonlocality; emphasis on its time aspect can be worded atemporality.’ Now (leaving apart the question whether algebraic entanglement of states gives *always* a geometric nonlocality) it seems, imo, interesting to point out (to myself) that the algebraic nonseparability has something to do with linearity, superposition, HUP, no-cloning, no-signaling, while geometric nonlocality has something to do with the apparent but anyway *uncontrollable* superluminality (A.Shimony), and the irreducible uncertainty about who performed, for first, the measurement on his entangled, but space-like separated, state. Maybe the right word is atemporality! This is also suggested, somehow, by Aravind in his paper about the topology of the GHZ state (Borromean & Brunnian rings, etc.) http://www.liv.ac.uk/~spmr02/rings/physics.html
and by N.Gisin and also Garisto (see in the ArXiv). And by Ne’eman (papers on geometry and non-locality) in ‘Foundations of Physics’, 16, (1986), page 361,page 361, and in ‘Proc. Nat. Acad. Sci. USA’, 80, (1983), p. 7051.
Well a classic example which may make this clearer is in communication complexity. Two parties, A and B, want to compute some joint function f(x,y) where A knows x and B knows y. We can ask how much they need to communicate in order to achieve this task. There exist function f(x,y) which, if the parties are allowed to communicate qubits instead of classical bits, then they can compute this function using exponentially fewer qubits. A physicist would say “The two parties can compute the function using k qubits of communication, yet it is impossible for the two parties to compute the joint function using k bits.” So a quantum protocol can do something which is impossible with a classical protocol. This is confusing because, it is certainly true that they can compute this joint function classically: they just need exponentially more classical bits. Often this leads to lose language as in your quote of Deutsch from the article.
It seems to me that if information is to be the glue unifying quantum theory with general relativity, then a better understanding of information and general relativity is needed. Until “gravity information theorists” exist in the same numbers that quantum information theorists do, I think there is too much of a conceptual divide between these fields. On the other hand, if geometry is to be the glue, then general relativity is certainly the leader in our understanding. 🙂
Ah. this is amazingly confusing. Thanks for the clarification though.
Wolfgang, I am not sure if information geometry has anything to do with gravitation. Information geometry is the study of probability as points on an appropriately defined Riemannian manifold. These ideas can be extended to quantum information via the use of complex manifolds. Fascinating stuff, but as far as I can tell nothing to do with gravitation.
> Wolfgang, I am not sure if information geometry has anything to do with gravitation.
Go to my webpage (just click on my name),
click on “The Statistical Mechanic”, scroll down until the second entry for 2005-10-01, read the two cited papers and be amazed.
Wolfgang: fascinating stuff. Thanks for the tip !