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原文摘要

If a quantum state is known only approximately, how much can the expectation of an observable change? We answer with the worst-case expectation over all states within a sandwiched Rényi divergence budget of a reference state, for every order $α\in [1/2,\infty]$. Convex duality gives this value as a one-dimensional optimization over a sandwiched Schatten norm, running from a fidelity risk at $α= 1/2$, through the entropic risk at $α= 1$, to a quantum conditional value-at-risk at $α= \infty$. The main result is a classicalization theorem: the quantum problem coincides with the classical one for a probability measure $ω_α$ on the spectral interval of the observable, constructed from Heinävaara's tracial joint spectral measure and given by Stahl's theorem at $α= 1$. The Hoeffding, Bernstein and Markov--Krein bounds of classical probability thereby apply to the quantum problem, and yield Pinsker-type inequalities with Bernstein-type refinements. The measure $ω_α$ is the Born measure only at $α= 1/2$; for $α> 1/2$, noncommutativity of the observable with the state strictly lowers its variance below the Born variance, and with it every variance-sensitive bound: a quantum deflation.