Browsing by Author "Plosker, Sarah"
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- ItemAn operator-valued Lyapunov theorem(2019) Plosker, Sarah; Ramsey, ChristopherWe generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak*-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).
- ItemBistochastic operators and quantum random variables(2022) Plosker, Sarah; Ramsey, ChristopherGiven a positive operator-valued measure ν acting on the Borel sets of a locally compact Hausdorff space X, with outcomes in the algebra B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H, one can consider ν-integrable functions X → B(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.
- ItemOn operator valued measures(2020) McLaren, Darian; Plosker, Sarah; Ramsey, ChristopherWe consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikod´ym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesque decomposition theorem and Johnson’s atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the “cleanness” relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.