Conditional quantum iteration from categorical traces

In order to describe conditional iteration in quantum systems, we consider categories where hom-sets have a partial summation based on an axiomatisation of uniform convergence. Such structures, similar to Haghverdi’s Unique Decomposition Categories (UDCs), allow for a number of fundamental constructions including the standard, or ‘particle-style’, categorical trace. We demonstrate that the category of continuous maps on Hilbert spaces falls within this fraqmework, and has a (partially defined) categorical trace based on iteration. This trace formula converges for unitary maps, but has no immediate physical interpretation. We then give a general construction that splits this trace into the composite of three maps: a canonical inclusion, a series of unitary operations, and a co-diagonal. We show that these unitary operations give a particle-style trace in a larger category (the convolution category), and demonstrate how the familiar (Elgot, Arbib-Manes) programming language interpretation of ‘conditional loops’ (via the standard trace over coproducts) gives a semantics of iteration conditioned on a purely quantum variable. Algorithms and physical interpretations are given.