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arXiv:2608.10191

Integration Theory for Completely positive Instruments: A Lyapunov-Type Theorem and Applications

Arghya Chongdar

Published Aug 10, 2026• 0 citations
arXivHTMLPDF

Abstract

We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).

DOI
10.48550/arxiv.2608.10191

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We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the C∗C^*C∗-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).
We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.
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For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
One of the most challenging open problems in quantum information theory is to clarify and quantify how entanglement behaves when part of an entangled state is sent through a quantum channel. Of central importance in the description of a quantum channel or completely positive map (CP-map) is the dual state associated to it. The present paper is a collection of well-known, less known and new results on quantum channels, presented in a unified way. We will show how this dual state induces nice characterizations of the extremal maps of the convex set of CP-maps, and how normal forms for states defined on a Hilbert space with a tensor product structure lead to interesting parameterizations of quantum channels.
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.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.