Recursion in Hilbert Space Semantics

Determine whether the category Hilb of Hilbert spaces and bounded linear maps supports a denotational interpretation of recursion, i.e., whether recursive definitions can be modeled categorically in Hilb (for example via suitable fixed-point or domain-theoretic structure).

Background

The paper contrasts inverse categories, which can model recursion via enrichment in directed complete partial orders, with categories based on Hilbert spaces, which lack the required enrichment (citing Heunen’s result). While inductive types can be interpreted in Hilb (via Barr’s theorem), the authors state that it remains unknown whether recursion itself can be interpreted in Hilbert spaces.

To address recursion for non-cartesian settings, the paper develops a guarded recursion framework using dagger categories rather than resolving the question for Hilbert spaces directly.