Dagger Categories of Tame Relations

Martin Strauss

Bart Jacobs

Within the context of an involutive monoidal category the notion of a comparison relation is identified. Instances are equality on sets, inequality on posets, orthogonality on orthomodular lattices, non-empty intersection on powersets, and inner product on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with "tame" relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation.

Jacobs, BPF Bart

This paper is a sequel to [19] and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance. 1

Arash E. Zaghi

We establish a rigorous connection between Relational Quantum Dynamics (RQD) and Categorical Quantum Mechanics (CQM) by formally showing that the category Q of quantum processes in RQD can be endowed with the structure of a dagger compact category. In particular, we demonstrate that Q satisfies the axioms of a dagger symmetric monoidal category with compact closure, mirroring the standard categorical formalism for quantum theory. Within this framework, key physical features of quantum mechanics, such as the transfer of awareness (information) in interactions, the no-cloning theorem, and the quantum teleportation protocol, are naturally expressed and interpreted. We define the categorical structures and illustrate the concepts with examples from quantum computing and quantum information. These examples highlight how RQD’s relational worldview aligns with CQM’s diagrammatic and algebraic characterization of quantum processes, thus providing a robust unified description. The results underscore that the RQD category Q not only has the formal properties of a dagger compact category but also that this structure provides clear insight into quantum informational tasks and no-go theorems, while incorporating the notion of observer-dependent reality in RQD via a functional mapping of awareness.

Hans Primas