Re: Quantum computation and categories

[John Baez <john.c.baez@gmail.com>]

Happy New Year!

Peter wrote:

Consider the following two categories:
>
> (a) the category of finite dimensional complex vector spaces and linear
> maps, and (b) the category of finite dimensional Hilbert spaces and linear
> maps.
>


> Specifically, take Toby's proposal, and consider two different objects A,B
> of (b) such that both A and B are two-dimensional Hilbert spaces. Let u:A->B
> be some non-unitary isomorphism.


Using "u" to stand for a non-unitary morphism!  Reminds me of the joke:

Teacher: Suppose p is a prime number...

Student: But what if it's not?

Teacher: Well then it wouldn't be called "p", now, would it!


> Then you can easily find an equivalence of categories which identifies both
> A and B with the two-dimensional vector space C^2, and which identifies u
> with the identity morphism on C^2. At this point, you have not equipped the
> category (a) with anything useful, because it does not induce a notion
> of unitary map on C^2.
>

Okay, that's a nice argument.  I'm pretty sure Lurie gave me some similar
argument: take a dagger-category, try to transport the structure along an
equivalence of categories, and get something unacceptable.

So it seems that, to define the extra structure of Hilbert spaces (on top of
> vector spaces), one needs at least one "evil" concept, be it that of unitary
> maps or the dagger structure.
>

 If this is really true (and I think it is), we're pushed towards Mark
Weber's idea: dagger-categories are not best thought of as categories but
rather something new, based on graphs-with-involution instead of graphs.

Best,
jb


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]