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arXiv · 1801.03109

An operator-valued Lyapunov theorem

Sarah Plosker·Christopher Ramsey
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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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Sarah Plosker, Christopher Ramsey. 2018-01-09. An operator-valued Lyapunov theorem. https://arxiv.org/abs/1801.03109

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