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arXiv · 2608.10191

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

Arghya Chongdar
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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}).

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Arghya Chongdar. 2026-08-10. Integration Theory for Completely positive Instruments: A Lyapunov-Type Theorem and Applications. https://arxiv.org/abs/2608.10191

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