Re: Quantum computation and categories

[Peter Selinger <selinger@mathstat.dal.ca>]

Several people attempted to give a "non-evil" definition of a dagger
category. Not much of this makes sense.

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.

Clearly, they are equivalent categories. They have the same morphisms!
Yet everyone knows that Hilbert spaces and complex vector spaces are
not the same. For example, one can define unitary morphisms
w.r.t. Hilbert spaces, but not w.r.t. complex vector spaces. The
concept of "unitary" is itself "evil", because it is not preserved
under isomorphism of objects in the category (b)!

So whatever extra structure the category (b) has, which allows a
definition of "unitary", must be evil. Transporting this along
equivalences does not make any sense whatsoever.

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. 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.

It is tempting to say that what is wrong with the category (b) is that
the morphisms don't accurately reflect the structure of the spaces.
Perhaps one would prefer to equip the category of finite dimensional
Hilbert spaces with unitary maps. Or with self-adjoint maps. Or with
isometries. Or with positive maps. The fact that there are so many
possible choices, and neither is strong enough to express all the
others internally, shows that this is not a good solution. One nice
feature of the dagger structure is that it does allow all of the above
to be expressed internally. So one gets lots of "evils" for the price
of one!

A sensible thing to do is to consider the category (b) of finite
dimensional Hilbert spaces with linear maps, to be also *equipped*, as
extra structure, with a distinguished lluf subcategory of isomorphisms
(the "unitary") ones. There is a natural notion of equivalence between
such categories-with-distinguished-subcategory (in particular, where
each component of the natural isomorphisms FG -> id and GF -> id is
required to lie in the subcategory). One can define a version of the
dagger structure for such categories-with-distinguished-subcategory
(in addition to the usual dagger axioms, one also must require that
the notion of "unitary" induced by the dagger structure coincides with
the distinguished subcategory that is a priori given).

Observe that the dagger structure can be transported along such
equivalence of categories-with-distinguished-subcategory. So the
dagger definition is non-evil on categories-with-distinguished-subcategory.

Unfortunately, the concept of a "distinguished subcategory" is itself
evil, if one does not require the subcategory to contain all
isomorphisms of the original category.

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.

-- Peter

Toby Bartels wrote:
>
> John Baez wrote in part:
>
> >A dagger-category is a category C with a functor
> >F: C -> C^{op}
> >which is the identity on objects and has F^2 = 1.
>
> >Category theorists will note that the above definition is "evil", in the
> >technical sense of that term:
> >http://ncatlab.org/nlab/show/evil
> >Namely, it imposes equations between objects, so we cannot transport a
> >dagger-category structure along an equivalence of categories.
>
> >Often evil concepts (like the concept of "strict monoidal category") have
> >non-evil counterparts (like the concept of "monoidal category").  But in
> >this particular case I know no way to express the idea without equations
> >between objects.  Both Hilb and nCob are dagger-categories.  This fact is
> >important.  Try saying it in a non-evil way!
>
> By default, there is a non-evil way to say it:
>
> Given a category C, a _non-evil dagger-category structure_ on C
> consists of a dagger-category C' and an equivalence F: C -> C' of categories.
>
> So one question is whether there is a less long-winded way to say that.
> Another question (which logically comes before the first question)
> is what is the right notion of equivalence of such structures.
>
>
> --Toby
>




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