Re: Quantum computation and categories

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

John Baez wrote in part:

>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.

Even in quantum theory, the obvious isomorphisms --that is,
the notions of how one Hilbert space may be equivalent to another--
are invertible linear isometries, equivalently the unitary maps.

To know what Hilbert spaces really "are", we only need to understand
the groupoid of Hilbert spaces, which has ~unitary~ maps as morphisms.
But we don't stop there; we look for a more interesting or useful category
whose underlying groupoid (in an appropriate sense) is this groupoid.
We could take the category whose morphisms are short linear maps;
then the invertible morphisms are precisely the unitary maps.
Or we could take the dagger category whose morphisms are
bounded linear maps and with the usual adjoint as the dagger;
the appropriate underlying groupoid in this context consists
not of all invertible morphisms but only of those morphisms
whose daggers are their inverses, which again are the unitary maps.
(I leave open the problem of defining an appropriate structure
on the category whose morphisms are all densely defined linear maps;
in fact, I'm not even sure whether this is even a category.
But this reduces to the previous case if we restrict to finite dimensions.)

We might instead take the category whose morphisms are bounded linear maps,
with no additional structure; but this gives us the ~wrong~ isomorphisms.
So going only half way is no good here.


--Toby


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