2020/04/30 by Sarah Plosker, Plosker, Sarah, Christopher Ramsey +1
Mathematics · #46B22 #46C50 #46G10 #47G10 #81P15 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2005.00005
openalex publication_date 2020/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Given a positive operator-valued measure ν acting on the Borel sets of a locally compact Hausdorff space X, with outcomes in the algebra \mathcal B(\mathcal H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space \mathcal H, one can consider ν-integrable functions X→ \mathcal B(\mathcal H) that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.