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

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 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 et al(douglus_plosker_ramsey_povmintegration₁).