The quantum instrument monad

• URL: http://arxiv.org/abs/2606.27805v1
• Date: Fri, 26 Jun 2026 07:45:03 GMT
• Title: The quantum instrument monad
• Abstract summary: We introduce the quantum instrument monad $mathcalI_mathcalA$.<n>It can be thought of as a noncommutative generalization of the state monad.<n>The construction of the measure-theoretic version is based on a new notion of integral of a quantum-operation-valued function against a state-valued measure.
• Score: 2.690502103971799
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Monads are a ubiquitous structure in functional programming used for modelling computational effects. For example, the state monad models the effect of a computation interacting with a memory system. Here we introduce the quantum instrument monad $\mathcal{I}_\mathcal{A}$, which models the effect of a computation interacting with a quantum system with algebra of observables $\mathcal{A}$. It can be thought of as a noncommutative generalization of the state monad. We construct this quantum instrument monad in two versions: a finitary version on the category of sets and a measure-theoretic version on the category of measurable spaces (the latter under the assumption that $\mathcal{A}$ is a type I von Neumann algebra with separable predual). Both versions are strong monads. The construction of the measure-theoretic version is based on a new notion of integral of a quantum-operation-valued function against a state-valued measure.

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