Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
They form a topological group what says it all.
Ian, automorphisms of any object form a group, the automorphism group of that object, and if you like you can regard this group as a 1-object category. This is described at delooping.
So the answer to your question is "yes", but it's not a particularly deep answer. It's an obvious yes.
There may be some diddling that needs to be done here. A priori a unitary is a (particular sort of) morphism from a Hilbert space to itself (in the category Hilb). A quantum channel is a (particular sort of) morphism from the algebra to itself (in some category of algebras, maybe ). The unitaries of a Hilbert space act on the endomorphisms by conjugation. This means we can map the unitaries into the quantum channels, but may not be one-to-one. I don't know off the top of my head if all maps from to itself (of the appropriate sort) are in the image of this map, but from what you say I suspect not. The partial overlap you envisage may in fact be that one is wholly contained in the other.
Also note that all this can be generalised to involve maps from one Hilbert space to another.
One thing I might suggest is that we have a page called Birkhoff's theorem, or at least a page containing it. Also some detail at quantum channels at what it means for a quantum channel to obey Birkhoff's thoerem. Then we can nut out what you really need here.
Is a quantum channel the (topological) group algebra of a unitarizable group (admits a faithful representation by a subgroup of the unitary group)? That is, instead of considering "almost everywhere vanishing" functions (the group ring), consider continuous functions with compact support (the group algebra of a topological group).
A quantum channel is a (particular sort of) morphism from the algebra End(H) to itself (in some category of algebras
Wait, a quantum channel is not in general an algebra homomorphism. Just a linear map.
Is a quantum channel the (topological) group algebra of a unitarizable group
Wait, a quantum channel is not an algebra, just a certain linear map.
I dunno, I guess I read wrong.
The collection of all endo-quantum channels forms a monoid, clearly, but not even a ring, because the sum of two linear maps underlying a quantum would not be trace preserving. So also this is not an algebra.
Unless I am missing something, of course.
But I am puzzled: don't we all agree now on the very simple definition of what a quantum channels is the way I wrote it at quantum channel? Let me know what is unclear. Maybe I am mixed up.
Edit: Never thee mind.
my original intent was to have the objects be the vector spaces. I honestly don't know if that really makes a difference or not.
it makes no difference: your original intent is completely realized here.
It makes no difference what you name the objects. There is one object per natural number . And the hom-set between the object labeled and the object labeled is a subset of the space of linear maps
You can -- and probably should -- think of the object labeled n as being the vector space .
Just that morphisms between these vector spaces are not linear maps between them, but linear maps between their vector spaces of endomorphisms.
Does that help?
My only potential concern (and I need to check on this) is that I could see different channels, both with the same value for n \in \mathbb{N}, having very different properties.
Hm? Of course they will have different properties. Otherwise this would be a bit boring. I don't understand: what's your concern?
Your statement sounds a bit as if you would also say: "I am concerned that there are linear maps from a vector space to itself that may have very different properties."
Yes. I'm curious about the channels which are their own inverses. It raises the question of channels which return the original information after passing it through n times... And in a more asymptotic mindset, for the (countable) inf. dim. Hilbert space case, how about channels which, for any small , there is an N such that for all we are within (in some measure of information) of the original state?
Also, how is the 'environment' coded? By coupling I assume you mean that one tensors the output with something, and there is a channel from this tensor product back to the original state, such that there is a composite (somehow) that is the identity?
Fixed.
I envisioned reversible channels were in a category whose arrows were isomorphisms while the non-reversible kind weren't, but all are clearly part the larger category QChan as Urs defined it.
that is correct.
other quantum- and relativity-related things
Sometimes it's not so much as whether things can be described using categories (since we think everything mathematical can be described using categories, sometimes in multiple ways) but whether that adds to the understanding. The thought of a category as encoding a process or information flow is one that is helpful and useful in both QM (Coecke and collaborators) and in computer science. Describing spacetime as having the structure of a poset can be useful when describing nets of observables in algebraic quantum field theory (what Urs calls functorial QFT edit: no he doesn't, see next comment). But beware there are some things that can be written in category theoretic language which essentially is just a replacement of words, and the structure etc when written as a category is exactly the same as before with no added value. I think quantum channels (especially when the classical information and the environmental interactions are included) do have some interest, though.
@Urs,
my mistake :)
<div>
I realized I never fully responded to your other post.<br/><br/><blockquote> Yes. I'm curious about the channels which are their own inverses. It raises the question of channels which return the original information after passing it through n times...</blockquote><br/><br/>Yes, indeed. And I had a hunch this was potentially related to one of the proposed extensions of Birkhoff's theorem to the quantum domain partly based on the fact that both the tensor product of copies of a channel (which is equivalent to executing copies simultaneously) and compositions of a channel (which is equivalent to repeatedly feeding the output back into the input) happen to both be bilinear operations (I know that's a really vague and probably unsatisfactory response, but I'm still mulling it over).<br/><br/>I'm not sure about the asymptotic case. I'll have to go back and look at the literature again.<br/><br/>Regarding the environment, it is just tensored with the output.
</div>
So what is the environment coded as?
I've not seen the partial trace before.
I'm not too surprised (from a mathematical viewpoint) that the density matrix depends on the map, and in fact makes things interesting. The challenge is then to figure out the rules which govern this.
I suggest that we continue this discussion at a page on the nlab, say quantum channel, in a new section about environmental interactions.
1 to 28 of 28