We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.

中文翻译：

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n 个任意观测值的 Wigner 分布

我们研究了 Wigner 函数对 Hermitian 算子的任意元组的推广。我们表明，对于 Hermitian 算子 A1、…、An 的任何集合和任何量子态，在 Rn 上存在唯一的联合分布，其性质是 Ak 的所有线性组合的边际与其对应的量子重合。换句话说，我们考虑所有线性组合的精确量子概率分布的逆 Radon 变换。我们称其为 Wigner 分布，因为对于位置和动量，该属性定义了标准的 Wigner 函数。我们讨论了在有限维系统中的应用，建立了许多基本性质，并通过例子来说明这些。属性包括支撑、奇点的位置、正性、对称群下的行为、