Re: Quantum computation and categories

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

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


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