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arXiv:2606.27805

The quantum instrument monad

Tobias Fritz

cs.LOcs.PLmath.CTquant-phSubmitted Jun 26, 2026• Announced Jun 29, 2026• 0 citations

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.

Comments: 28 pages. Independent work by Booth, Leichtle, Rice and Worrall develops a closely related construction in the Heisenberg picture. The two works provide complementary Schrödinger- and Heisenberg-picture formulations

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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.

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.

Many production quantum programming languages represent hybrid quantum computations by extending a classical base language with a quantum effect, where qubits are addressed by reference, and quantum operations are understood to mutate some external quantum state. However, the semantics of this view of quantum computation remains underdeveloped, especially when the language allows mid-circuit measurements and non-termination. In this work, we provide a general method for building denotational semantics for such languages, by defining the quantum orchestra monad, which precisely captures this style of quantum effect. The monad has a concrete presentation, being based on the formalism of quantum instruments, a common tool in quantum information theory for capturing the action of a quantum process along with its classical outcomes. It acts on the category DCPO, and so enables the interpretation of divergent hybrid programs. The quantum orchestra monad serves as a natural extension of both the classical state monad and the probabilistic powerdomain monad. We investigate some of the subtleties present when trying to naïvely extend these definitions to the quantum non-commutative case.

The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space

\mathcal{H}

. In terms of the Hilbert lattice

\mathcal{L}

of closed linear subspaces of

\mathcal{H}

the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category

\mathbf{QCB}_0

which is equivalent to the category

\mathbf{AdmRep}

of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.