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Magic states are key ingredients in schemes to realize universalfault-tolerant quantum computation. Theories of magic states attempt toquantify this computational element via monotones and determine how thesestates may be efficiently transformed into useful forms. Here, we develop astatistical mechanical framework based on majorization to describe Wignernegative magic states for qudits of odd prime dimension processed underClifford circuits. We show that majorization allows us to both quantifydisorder in the Wigner representation and derive upper bounds for magicdistillation. These bounds are shown to be tighter than other bounds, such asfrom mana and thauma, and can be used to incorporate hardware physics, such astemperature dependence and system Hamiltonians. We also show that a subset ofsingle-shot R\'{e}nyi entropies remain well-defined on quasi-distributions, arefully meaningful in terms of data processing and can acquire negative valuesthat signal magic. We find that the mana of a magic state is the measure ofdivergence of these R\'{e}nyi entropies as one approaches the Shannon entropyfor Wigner distributions, and discuss how distillation lower bounds could beobtained in this setting. This use of majorization for quasi-distributionscould find application in other studies of non-classicality, and raises novelquestions in the context of classical statistical mechanics.

Constraints on magic state protocols from the statistical mechanics of Wigner negativity