Universal Properties of Partial Quantum Maps

Abstract

We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be applied to any dagger rig category, is described in three steps, each associated with their own universal property, and draws on results from dilation theory in finite dimension. In this way, we explicitly construct the category that captures hybrid quantum/classical computation with possible nontermination from the category of its reversible foundations. We discuss how this construction can be used in the design and semantics of quantum programming languages.

Axioms for the category of Hilbert spaces and linear contractions

“…Quantum systems may only evolve along unitary linear maps. Taking quantum measurement into account allows the larger class of linear contractions [1, 26]. But no quantum‐theoretical process can be described by a bounded linear map that is not a contraction.…”