The question of how to characterize disorder arises in a wide variety of settings, including information theory, economics, and quantum thermodynamics. The theory of majorization provides an elegant answer, playing a central role in these fields. However, the existing majorization framework is inapplicable to functions that both take negative values and are defined on infinite spaces. Yet such functions, in the form of quasiprobability distributions, are ubiquitous in fields such as quantum optics, signal analysis, and bosonic quantum computation. Here we develop a notion of majorization that is applicable to such functions, proving that it admits four equivalent characterizations that naturally reduce to the finite case, thereby generalizing a seminal theorem by Hardy, Littlewood, and Pólya. Moreover, we extend this equivalence to the setting where majorization is considered relative to an arbitrary positive distribution. We give several applications of our results in the context of quantum resource theories. These include deriving families of resource monotones and constraining quantum state conversions. We analytically and numerically study examples using the Wigner and Husimi functions, which feature prominently in quantum optics. Our results provide a comprehensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces. Quasiprobability distributions, crucial in quantum optics and computation, challenge existing majorization frameworks due to their negative values and infinite domains. Here, the authors establish a novel majorization concept for such functions, offering four equivalent characterizations and extending applications to quantum resource theories, enhancing our understanding of quantum state transformations.

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Nature

A new majorization framework is developed for quasiprobability distributions — functions that take negative values over infinite domains — which are central to quantum optics, signal analysis, and bosonic quantum computation. The work proves four equivalent characterizations of this generalized majorization, extending the classical Hardy-Littlewood-Pólya theorem to infinite measure spaces. Applications include deriving resource monotones and constraining quantum state conversions within quantum resource theories, with analytical and numerical examples using Wigner and Husimi functions.

Majorization theory for quasiprobabilities