C*-Extreme Quantum Instruments; Completion and Disintegration of Completely Positive Maps

Abstract

This thesis develops a comprehensive operator-algebraic framework for the study of completely positive (CP) instruments, with contributions spanning convexity theory, integration theory, completion problems, and disintegration theory. We begin by laying the foundational groundwork in the theory of C*-algebras and von Neumann algebras, introducing key structures such as CP maps, positive operator-valued measures (POVMs), and CP instruments, along with their dilation-theoretic properties. Pure and decomposable instruments are characterized via minimal bi-dilations, and CP instruments are realized as bivariate maps, providing a rigorous quantum analogue of classical joint measures. The thesis then investigates convexity-theoretic aspects of instruments. A structural characterization of C*-extreme unital completely positive (UCP) instruments on finite-dimensional Hilbert spaces is established, employing methods from the theory of nest algebras. The interplay between C*-extremality and the marginals of an instrument is studied, yielding results on the spectral nature of POVM marginals and the unique determination of an instrument from a single C*-extreme marginal. A systematic integration theory with respect to CP instruments is then developed, inspired by Bartle's classical vector integration framework. This culminates in a CP-instrument correspondence theorem on compact Hausdorff spaces and, notably, a Krein-Milman type theorem for CP instruments on separable C*-algebras — a result not previously available in the literature. The thesis further addresses the CP completion problem — the extension of partially defined linear maps to fully CP maps on C*-algebras — establishing the existence and uniqueness of minimal CP completions, and generalizing a result of Parzygnat and Russo on almost-everywhere identity maps to the full generality of von Neumann algebras. Finally, the theory of non-commutative disintegration is developed, connecting classical disintegration to the existence of left inverses for CP maps. Structural results for left-invertible normal CP maps on B(H) are obtained, and existence and uniqueness of disintegrations are established in the infinite-dimensional setting.