From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5466 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Quantum computation and categories Date: Sun, 3 Jan 2010 16:38:11 -0800 Message-ID: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1262574707 23851 80.91.229.12 (4 Jan 2010 03:11:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Jan 2010 03:11:47 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon Jan 04 04:11:40 2010 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 1NRdM3-0007OH-Ti for gsmc-categories@m.gmane.org; Mon, 04 Jan 2010 04:11:40 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NRcuG-0002VJ-OU for categories-list@mta.ca; Sun, 03 Jan 2010 22:42:56 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5466 Archived-At: Fred E.J. Linton wrote: > Peter Selinger offered the thought that, considering > > > ... the category of finite dimensional complex > > vector spaces vs. the category of finite dimensional Hilbert spaces. > > They are equivalent ... > > Hmmm ... you mean just *any* linear transformation is allowed between two > Hilbert spaces? > In applications to quantum mechanics people really want to work with both unitary and self-adjoint operators, and often others as well. So they work with the category of finite-dimensional Hilbert spaces and *all* linear maps between these. As a mere category this is equivalent to the category of finite-dimensional vector spaces - so to understand the "Hilbertness" of Hilbert spaces, they introduce a dagger structure as well. (The infinite-dimensional case would introduce extra wrinkles, like unbounded self-adjoint operators. It's possible that only after we treat this case correctly can we declare that we know what's going on. Perhaps trying to treat both unitary and self-adjoint operators as morphisms in the same category is simply a bad idea. There are a lot of options worth exploring.) If so, I'm not so sure my Hilbert spaces are the same as yours :-) . > Indeed! If you treat Hilbert spaces as "sets with structure", the obvious morphisms are isometries - inner-product-preserving linear operators. But in quantum theory, Hilbert spaces are being used for something quite different. And so there's a struggle going on to understand this. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]