Re: quantum information and foundation
Dear Andre, Thorsten, Dusko, and others, Andre Joyal wrote:
Quantum information science is also quite speculative:
http://en.wikipedia.org/wiki/Quantum_Information_Science http://en.wikipedia.org/wiki/Topological_quantum_computer
It depends whether one is talking about: (1) having a quantum computer in the shops (2) theoretical discovery and experimental verification of new physical phenomena inspired by approaching nature in information-theoretic terms. While the first is indeed pure speculation, the second is a fact, with many recently discovered physical phenomena, some of which are embodied in terms of computational models, having effectively been established in the lab. Well-known examples are quantum teleportation and quantum key exchange. To mention one example of a phenomenon embodied in terms of computational model: the ability to universally alter the state of quantum systems by only relying on observations (= the measurement-based quantum computational model). Actually, certain guises of quantum information technology are effectively available for purchase at: ID quantique: http://www.idquantique.com/ MagiQ: http://www.magiqtech.com/MagiQ/Home.html Smart Quantum: http://www.smartquantum.com/SmartQuantum.html These three companies are not at all controversial, as opposed to for example D-Wave. There must be well over 1000 researchers active in the area which has its `own wikipedia': http://www.quantiki.org/wiki/index.php/Main_Page The general expectation would be that it are the quantum communication protocols which will be the first transitions to mainstream technology, and these may become components within some hybrid information processing device. Andre Joyal wrote:
But the mystery of quantum physics lies elsewere: the extraction of a probability distribution from the complex values of a wave function.
Thorsten Altenkirch wrote:
I agree that the big question in quantum theory is the "measurement problem".
The measurement-based quantum computational model is interesting in that it considers what for a long time was the most controversial ingredient of quantum theory, as the main processing resource: von Neumann's projection postulate which describes how the state changes under observation. These changes under observations of typically highly entangled states can be conveniently modeled by certain interacting Frobenius algebras in monoidal categories: http://arxiv.org/abs/0906.4725 http://arxiv.org/abs/0902.0500 I don't see any speculation here, just a convenient manner of representing physical phenomena which effectively have been observed in the lab, by using structures which are considered as category-theoretic. A software package to help with this is also under development: http://web.comlab.ox.ac.uk/people/Aleks.Kissinger/projects.html http://dream.inf.ed.ac.uk/projects/quantomatic/ In this context, recently Ross Duncan and Simon Perdrix solved an open problem in the area of measurement-based quantum computing, which has to do with guarantying a deterministic answer for certain sequences of measurements, and the formulation of the answer crucially relies on the Frobenius algebras. (their paper is forthcoming) Dusko Pavlovic wrote:
most physicists would probably say that they are happy with hilbert spaces. but many of them (albeit mostly theoreticians) ar enot.
In fact, it are the experimentalists which tend to get quite excited about the use of graphical languages to describe quantum phenomena since these are more `operational' than the usual Hilbert space treatments. Theoretcians have a harder time to denounce the things to which they are used, except when you are called John von Neumann and you crafted the Hilbert space quantum mechanical formalism a few years earlier. Best wishes for the new year, Bob. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Bob, Thorsten and Dusko, I thank you for expressing frankly your position. Thorsten wrote:
I have to admit that I am quite ignorant about many of the areas mentioned in the previous email. On the other hand developments in Computer Science I know about don't seem to feature. Maybe they are too mundane for Mathematicians.
Dusko wrote:
is it just my impression, or are category theorists a little more sceptical about the value of applications than most mathematical communities? they seem to seek a recognition that categories are useful across mathematics, but then hesitate to recognize the depth and value of the applications in the other areas. --- can it be that we suffer from a superiority complex of some sort?
Bob wrote:
In this context there is the highly unfortunate fact that there are certain quite prominent people in the category theory community who think that any deviation from treating category as a branch of pure mathematics and pure mathematics only is a bad thing!
You seems to suggest that this is a debate between pure and applied category theorists. I disagree to the extend that "quantum foundation" and "quantum information" are very speculative subjects. The "Foundational Question Institute", http://www.fqxi.org/ which is known to support speculative research projects exclusively, is funding a project on Quantum Foundation by Bob Coecke: http://www.fqxi.org/grants/large/awardees/view/__details/2008/coecke It has funded a project called "Topos Quantum Theory" by Christopher Isham http://www.fqxi.org/grants/large/awardees/view/__details/2006/isham It is funding a project "Categorifying Fundamental Physics" by John Baez: http://www.fqxi.org/grants/large/awardees/view/__details/2008/baez Physics is in bad shape today according to Lee Smolin: http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/ His main critic is that string theory has lost contact with experiments. It has become an academically driven discipline. Maybe we should stop calling it physics. Of course, it can be interesting mathematically. In mathematics, the word "quantum" is often used as a prefix to express some vague connection to quantum physics, like non-commutative algebras and Feynman diagrams. By itself it is no proof that the named notion is fitting something in the natural world. There are quantum groups, quantum algebras, quantum Grassmanians, quantum planes, quantum bundles, quantum Schubert cells, quantum cohomology theories, quantum fields, quantum Yan-Baxter operators, etc. The theory of quantum groups is mathematically very interesting but it has no applications that I know to real quantum physics: http://en.wikipedia.org/wiki/Quantum_group I have a Phd student working on quantum quasi-shuffle algebras and he needs not to know about quantum physics because it is irrelevant. The notion of dagger compact closed category is interesting and purely mathematical, like the notion of quantum group. Quantum information science is also quite speculative: http://en.wikipedia.org/wiki/Quantum_Information_Science http://en.wikipedia.org/wiki/Topological_quantum_computer Again, there is nothing wrong with highly speculative research. So the present debate is not about real applications of category theory. Bob wrote:
What the quantum information `hype' has done is injected some new blood in foundations of quantum mechanics research, an area which for several reasons had been suffocated by by the end of the previous century, despite the universal discomfort of the physics community with quantum mechanics. (a typical slogan which reflects this is: ``don't ask questions just compute'') One would expect that this surge of quantum foundations, which meanwhile has led to many novel ideas, approaches, and radically different manners to think about physics in general, will ultimately lead to new mathematics. Moreover, the natural guise of many of these new ideas is within category theory, a message that some including myself have been trying to pass on within the foundations of physics communitee, with moderate success.
Feynam introduced his diagram as a method for computing the solutions of QED field equations. It is essentially a technique for enumerating the terms arising in perturbation theory. The method was extended to all physical fields and Penrose understood the connection between the diagrams and tensor calculus. The geometry of tensor calculus is just an abstraction of this connection. Feynman diagrams are very useful in physics and mathematics. But the mystery of quantum physics lies elsewere: the extraction of a probability distribution from the complex values of a wave function. I dont think that a categorical formalism based on Feynman diagrams is very different from what the physicists are currently doing. This maybe why your formalism is having a moderate success among the physicists. Of course, a good formalism can stimulate new developements. But it should not be presented as radically new if it is not. Too much hype may backfire. It is not good for the reputation of category theory. Happy New Year to all! Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Mon, Dec 28, 2009 at 6:54 PM, Joyal, André <joyal.andre@uqam.ca> wrote:
Physics is in bad shape today according to Lee Smolin:
http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/
His main critic is that string theory has lost contact with experiments. It has become an academically driven discipline. Maybe we should stop calling it physics. Of course, it can be interesting mathematically.
From that perspective string theory strongly deserves to be studied by
I would like to expand on this remark, and point out an application of (higher) category theory that might deserve more attention from mathematicians. First a remark concerning the detachment of string theory from experiment: much of theoretical physics, not just string theory, is far remote from experiment, but -- in principle -- for a good reason: if experiment shows that a certain incarnation of mathematical structure X is relevant for the description of the physical world, then for understanding it well we ought to study also all other incarnations of structure X, even if they are not (yet) known to be relevant for the description of the world themselves. As a simple example: not all solutions of Einstein's equations describe anything in the real world. But we want theoretical physicists to understand as many as possible of them: while some particular cosmological model (say one with closed timelike geodesics) may look utterly irrelevant for the description of the real world (given the present state of experimental knowledge!), it is the understanding of the collection of all such models and their interrelation that helps with understanding the particular one that does describe the real world. This idea, that we may study a theory in terms of the collection of its models, should resonate with category theorists. theoretical physicsists, even in the absence of experimental evidence: the string perturbation series is a conceptually compelling variation of Feynman's celebrated sum over correlators of a 1d QFT. Every theoretical physicist worth his or her money should feel an itch to explore the analogous sums over correlators of 2d QFTs. And that's what (perturbative) string theory is. http://ncatlab.org/nlab/show/string+theory And indeed, the above idea that for understanding one model it helps to understand all its variations, is at work here, too: studying the string perturbation series has led to a better understanding of Feynman's perturbation series, since a few years quite spectacularly resulting in a previously undreamed of understanding of the higher loop Feynman terms in supergravity theories. The fact that the discovery of many other suggestive aspects of the string perturbation series made a whole community become so excited about it that they threw some care and scientific discipline in the wind is a problem, but one of the sociology of science, not a fault of the topic. The reason why I feel saying all this is worthwhile on a mailing list devoted to category theory, is that a closer look shows that the mathematical structures involved in string theory are not only an impressive source of examples of applications of higher category theory, but in some cases even their archetypical motivational examples. The cobordism hypothesis/theorem http://ncatlab.org/nlab/show/cobordism+hypothesis is arguably comparatively pivotal for higher category theory as, say, the Yoneda lemma is for ordinary category theory. (I really think it is.) With that in mind, it should not be forgotten that both its roots in the ideas of Witten, Atiyah and Segal, as well as its present rather impressive applications in the work of Freed-Hopkins-Lurie-Teleman http://arxiv.org/abs/0905.0731 are situated in the conceptual framework that was opened by the step from the Feynman perturbation series to string theory: as John Baez mentioned in a previous message, cobordism representations are being speculated to encode quantized general relativity, but that speculation should not make us forget that what made theoretical physicists eventually pass from the study of quantum field theories defined on Minkowski space or similar, to "full" quantum field theories defined on all possible cobordisms was the idea that the Feynman perturbation series ought to have a generalization from a sum over graphs to a sum over cobordisms of higher dimension: conformal field theory used to be studied on R^2 for years until string theory opened the perspective that a CFT ought to be defined on general surfaces. Today the classification of such full 2dCFT -- the representation theory of 2-dimensional conformal cobordism categories -- is an impressive result in the theory of modular tensor categories. http://golem.ph.utexas.edu/string/archives/000813.html Indeed, it seems to me that the most substantial conceptual progress on the grand perspective exhibited by the passage to the string perturbation series has recently come not out of the physics departments (which seem to be curiously stuck with throwing insufficient formal tools at their grand targets), but out of the math departments, those math departsments where higher category theory has an influence in one way or other. In order to proliferate this observation, with AMS publishing we are currently preparing a book volume that is devoted to exhibiting aspects of the full story behind this claim. http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+... The text at that link may provide more details on the point that I am trying to make here. I can summarize this point maybe as follows: pure mathematicians and especially category theorists and higher category theorists should not be tricked by complaints such as voiced in Smolin's book into thinking that it is ill-advized to have a closer look at the mathematical structures to be found in string theory, well hidden under physicist's nonsense as they may be. On the contrary: much of what makes the present practice of string theory so tiresome is that the lively activity of the 1980s of mathematically inclined researchers looking into the mathematical structures of the theory has largely vanished, at least in the physics departments. The theory is much more interesting than the average talk of its current practicioners. And much deeper. One of the foremost powers of category theory is its ability to unravel hidden structures and make them become mathematically active. String theory is a vast reservoir of crucial (higher) categorical structures that is, while recently beginning to be investigated as such, largely like a huge bag of disjoint LEGO pieces which physicist dream of putting together to a grand edifice, but which is waiting for the higher category theorist to actually assemble it. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Prof. Joyal, 1. I agree with you that the hype about combinatorics of Feynman diagrams is, while important for constructing good practial theories and calculational methods, not appropriate target for understanding and changing the very foundations of quantum theory. 2. I disagree with you that quantum groups have no applications to real quantum physics. Surely, they do not change the very foundations of quantum theory, but do have numerous and significant applications to concrete models in quantum physics. Most of the significant applications are limited to the quantum groups at root of unity. They appear as symmetries of numerous integrable models, e.g. quantum spin chain models, and hidden symmetries of some conformal field theories to name the most well-understood applications. Harmonic analysis on quantum groups is important to calculate analytic expressions for correlation functions in some of the models, and the representation thoery at root of unity has a Kazhdan-Lusztig type correspondence in some cases to vertex operator algebra representations. This involves not a superficial but a very intricate picture. As a physicist I despise when people come with quantum and string terminology when not at least vaguely and indirectly appropriate, revelations by mathematician that they found the true meaning of some physical concepts and alike. A typical claim is of many mathematicians that vertex operator algebras are THE SAME as conformal field theories, while they feature just a part of the true story. I witnessed a talk by a young hot mathematician who gave an introduction that CFT as a discipline is a SUBSET of string theory. When I told him that CFT originated and is fruitful outside of string theory too (e.g. in study of critical phenomena in condensed matter physics), and thus should not be DEFINED subordinated to its particular hot and popular application, he started substantiating his claim waving hands that somebody has proved that "this and this is the same as that and that" (I am not paraphrasing but citing!!! what kind of psychology drives these young postdocs from Princeton-level hype places snowing the audience with misterious claims and referal to untouchable authorities whom they seen somewhere and half-understood ??). 3. As far as quantum computation and quantum information, the engineering boundaries of the field are not natural place of subject within physics and math. If one looks at the textbooks on quantum information more than half of the books are just standard material on quantum physics, not a different area. Topological quantum computation on the other hand, is more of topology, monoidal categories and QFT-type in its technology so it is already included in divisions listed. Various measures of coherence on the other hand in the literature are rather nonrigorous and somehow trivial variations are publiashable. I have been a referree 2 times and witnessed extremely content-free papers building the merit on 2-3 elementary and obvious observations which were claimed to have connections to algebaric geometry etc. while the authors were not being able to say anything nontrivial other than fancying about formal similarity in a polynomial describing some quantity. The other referree, from optical engineering has suggested the papers for publications as "significant" in J. Phys. A which accepted it against my recommendations. Baisng publications on hype and superficial remarks other than substantial content is a sign of an unhealthy standpoint of the community. I agree with John Baez that there is a healthy potential in quantum computing, but do not think that the area is well-defined, not subsumed to already listed areas of applications (like QFT), and would remark that it is overfunded for the present extent of true significant research. 4. It is not very important how we subdivide the applications of categories, but it is more important that we educate each other with aspects and overview of the subjects some of us are not specialized in but others can help. Awareness of possible applications amy help to bridge the gap between special areas and main focuses of current pure research. Thus while the lists like the one compiled in this discussion may be fun to mobilize a bit of cross-disciplinary discussion, more educative efforts and true discussions would do more. Zoran [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Bob Coecke -
Joyal, André -
Urs Schreiber -
zoran skoda