An operator-valued Lyapunov theorem |
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Authors: | Sarah Plosker Christopher Ramsey |
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Affiliation: | 1. Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada;2. Department of Mathematics and Statistics, MacEwan University, Edmonton, AB T5J 4S2, Canada |
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Abstract: | We 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). |
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Keywords: | Operator valued measure Quantum probability measure Atomic and nonatomic measures Lyapunov Theorem |
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