Axioms for the category of Hilbert spaces and linear contractions
Chris Heunen, Andre Kornell, Nesta van der Schaaf
TL;DR
This work provides a purely categorical characterisation of the category of Hilbert spaces with linear contractions by axioms for a dagger rig category with two monoidal structures. By localising at all nonzero scalars, the contraction category is completed to recover the full category of bounded linear maps, i.e. $\mathbf{Hilb}_{\mathcal{C}}$ with $\mathcal{C}=\mathbf{C}(I,I)$, an involutive field isomorphic to $\mathbb{R}$ or $\mathbb{C}$. The main theorem shows that any dagger rig category satisfying the presented axioms is equivalent to $\mathbf{Con}_{\mathbb{R}}$ or $\mathbf{Con}_{\mathbb{C}}$, with the completion yielding Hilbert spaces; conversely, such an equivalence implies the axioms. This establishes a robust, modular reconstruction blueprint for quantum-information-relevant categories from purely categorical properties, bridging Con, Hilb, and the broader landscape of categorical quantum mechanics.
Abstract
The category of Hilbert spaces and linear contractions is characterised by elementary categorical properties that do not refer to probabilities, complex numbers, norm, continuity, convexity, or dimension.