Composing Quantum Instruments
For quantum information theorists, this provides a rigorous categorical framework for composing quantum instruments, addressing a foundational gap in the theory.
The paper constructs a composition operation for quantum instruments with continuous outcomes, analogous to Markov kernel composition, by developing an integral of quantum channel-valued functions. This yields a monad whose Kleisli category is the category of quantum Markov kernels.
We study the composition of classically-controlled quantum instruments--the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.