Hilb is category which objects are Hilbert spaces and morphisms are bounded maps. It is dagger symmetric monoidal category. How can we ensure, that some arbitrary monoidal category is Hilb?


So you want some additional axioms for a symmetric monoidal category such that if $\mathbf C$ satisfies them then $\mathbf C\simeq\mathbf{Hilb}$?

This is an active area of research! People think of it as giving an operational definition of quantum theory. In this context the word "operational" means "in terms of processes and how they interact". Several such axiomatisations have been given, mostly by people associated with the QUIT group at the University of Pavia. For example these three papers (I'm not an expert here so I can't guarantee that those papers are the best examples and haven't been surpassed by more recent work)(EDIT: And this book).

Of course there's one easy way to axiomatise $\mathbf{Hilb}$; just add the axiom that $\mathbf C\simeq\mathbf{Hilb}$! But this is stupid. What we really want is some simple axioms that (after some work) turn out to imply that $\mathbf C\simeq\mathbf{Hilb}$. So axiomatisations can be judged by how much of the structure of $\mathbf{Hilb}$ they have to put in "by hand" in their axioms, compared to how much they can derive.

The papers I linked to above work in the framework of operational probabilistic theories (OPTs). An OPT can be seen as a symmetric monoidal category equipped with some extra structure (the exact relation between OPTs and SMCs is described in this nice paper by Sean Tull). In particular the definition of an OPT already requires that the scalars (i.e. maps $I\rightarrow I$) are precisely the real numbers in $[0,1]$, i.e. the probabilities*. In my own opinion I would say that this is quite a strong assumption, but you might disagree.

I believe that there is a paper coming out soon that tries to give axioms for $\mathbf{Hilb}$ purely in the framework of SMCs without using OPTs. If I remember correctly, John Selby will be one of the coauthors. I'll try to remember to update this answer when it comes out.

*Ah, this just made me realise that the papers aren't characterising $\mathbf{Hilb}$ but rather $\mathbf{CP}(\mathbf{Hilb})$, the category of Hilbert spaces and completely positive (and trace reducing) maps. This doesn't really matter since it's not too hard to characterise $\mathbf{Hilb}$ and $\mathbf{CP}(\mathbf{Hilb})$ in terms of each other.

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