From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5429 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: quantum information and foundation Date: Sun, 27 Dec 2009 23:14:23 +0000 (GMT) Message-ID: References: Reply-To: Dusko Pavlovic NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII; format=flowed X-Trace: ger.gmane.org 1262041810 29473 80.91.229.12 (28 Dec 2009 23:10:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 28 Dec 2009 23:10:10 +0000 (UTC) To: Original-X-From: categories@mta.ca Tue Dec 29 00:10:03 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NPOiv-0004rq-Rc for gsmc-categories@m.gmane.org; Tue, 29 Dec 2009 00:10:02 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NPOGq-0007AQ-Fb for categories-list@mta.ca; Mon, 28 Dec 2009 18:41:00 -0400 Content-ID: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5429 Archived-At: dear andre, first of all, i would like to thank you again for this invigorating thread. it seems that we are getting to some points that seem to be of general interest, so i'll add some comments. > 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. physicists and category theorists are certainly drawing very similar string diagrams. the *meaning* of these diagrams is, however, not completely identical. for physicists, string diagrams are a convenient shorthand for some constructions with hilbert spaces and operators. for category theorists, string diagrams represent constructions available in any category with enough structure. how important is this difference? more precisely, how important is it to go beyond hilbert spaces and look for some ***nonstandard models***? most physicists would probably say that they are happy with hilbert spaces. but many of them (albeit mostly theoreticians) ar enot. von neumann was very unhappy, and worked a lot to provide alternatives. and failed. but many people are thinking hard about "toy models" these days, capturing certain quantum phenomena and not other, generating some independence results, axiomatics etc. maybe category theory can help with this. (eg, bob coecke et al's recent work, as well as some bits that i have worked on, show that some crucial quantum phenomena, even entire quantum algorithms, can be represented using funny constructions with relations.) of course, my view of this may be biased, and nonstandard models of quantum mechanics may be irrelevant. but this is just one direction, showing a general way in which popping up from concrete sense into abstract nonsense may be a good thing. > Of course, a good formalism can stimulate > new developements. But it should not be presented as radically new if it > is not. To much hype might backfire, with bad consequence for the social > image of category theory. i cannot agree with this more. my first post in this thread was that maybe we should not advertise too much, but just make our tools available. ("nature will find the way" says the mathematician in jurassic park) > 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. oh but is that a bad thing? differential calculus was first physics, and then captured a lot of other things as well. and some of it did not reflect back into physics. as a computer scientist, i tend to think of quantum mechanics as a theory of a particular computational resource: **entanglement**. it seems to me that this concept raises fundamental worries for every computer scientist --- completely independent on its physical realisation. church's thesis said that computability was a very robust notion: whatever kind of a computer you take, you can compute the same. and for a while, it seemed that feasibility would be similar: there are various complexity classes, but they are all strictly subexponential with respect to each other. --- then came quantum algorithms with their "exponential beast", lurking from entanglement. now we know that computation happens in many models: on the internet, in a cell, distributed among the members of a mailing list. can some of them compute essenticall more than others? i don't know much about physics, but i cannot stay away from thinking about entanglement, and tensors, and string diagrams... with the very best wishes, -- dusko PS re **nonstandard models** again, i am wondering whether the hasegawa-hoffman-plotkin-selinger (HHPS) results, referred to by peter selinger, imply that there are no nonstandard models. the HHPS results say that a diagram commutes in a dagger-compact (resp compact, traced monoidal) category if and only if it holds in finitely dimensional hilbert (resp vector) spaces. so if a nonstandard model must be dagger compact, then anything validated in it must be validated in hilbert spaces? that would pretty much kill my nonstandard models, wouldn't it? i am not sure that i completely understand the HHPS results (so please correct me if i am wrong), but it does not seem to me that they provide anything like a representation theorem. a representation theorem, say for abelian categories, says something like: you give me a small abelian category AA, and i produce a ring R and an embedding AA--->Mod-R. in contrast, the HHPS theorems say: you give me a diagram D that commutes in a dagger-compact category, and i provide a field K such that that D also commutes in FHilb_K. so for every D, i need to construcat a new field K_D, right? well, this would provide an embedding of a dagger compact-category CC into FHilb_H for some field H only if there was a way to all fields K_D for all diagrams D that commute in CC into one big field H. how much hope is there for that? and even if i could do that, it would take some massage to embed FHilb_H into the standard model, consisting of *complex* hilbert spaces. so i still think that hilbert space may be a needlessly big place. 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