AI 中文总结
该研究基于沃尔什-哈达玛表示发展施密特规范非局域魔性的数学理论，证明其对任意1×N分块的精确性，推导相关分析性质，将其与离散调和分析关联，为非局域魔性的研究提供了新框架。
AI 中文摘要
量子态的完全非稳定子性（魔性）包含局域的、依赖基的贡献，而基于对局域幺正的极小化的非局域公式化，可分离出与真正非局域关联相关的分量。在该框架内，非局域魔性的施密特规范公式化，建立了真正非局域非稳定子关联与量子多体态的纠缠谱之间的直接联系。基于我们配套Letter中引入的精确沃尔什-哈达玛表示，我们发展了与该公式化相关的数学理论。我们将该形式主义扩展到任意分块，证明了施密特规范对任意1×N分块的精确性，并推导了施密特规范非局域魔性的一般分析性质，包括纠缠界、选择定则以及与归一化沃尔什谱矩的精确关系。该表示还允许将非局域魔性解释为沃尔什纠缠谱的逆参与率，从而将其与沃尔什空间中的集中度度量关联起来。这些结果表明，沃尔什-哈达玛表示揭示了隐藏在原始谱公式化中的潜在离散调和结构，它不仅提供了施密特规范非局域魔性的等价表达式，还为其分析研究提供了自然的数学框架，将该理论置于更广泛的离散调和分析背景中。
英文摘要
While the full non-stabilizerness (magic) of a quantum state contains local, basis-dependent contributions, a non-local formulation based on a minimization over local unitaries isolates the component associated with genuinely non-local correlations. Within such a framework, the Schmidt-gauge formulation of non-local magic provides a direct connection between genuinely non-local non-stabilizer correlations and the entanglement spectrum of quantum many-body states. Building on the exact Walsh--Hadamard representation introduced in our accompanying Letter, we develop the mathematical theory associated with this formulation. We extend the formalism to arbitrary bipartitions, prove the exactness of the Schmidt gauge for arbitrary $1\times N$ bipartitions, and derive general analytical properties of Schmidt-gauge non-local magic, including entanglement bounds, selection rules, and exact relations with the moments of the normalized Walsh spectrum. This representation also provides an interpretation of Schmidt-gauge non-local magic as a logarithmic inverse participation ratio normalized by a universal harmonic baseline, thereby relating it to excess delocalization in Walsh space. These results demonstrate that the Walsh--Hadamard representation reveals an underlying discrete harmonic structure that is hidden in the original spectral formulation and provides considerably more than an equivalent expression for Schmidt-gauge non-local magic. Rather, it furnishes the natural mathematical framework for its analytical investigation, placing the theory within the broader context of discrete harmonic analysis.
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