AI 中文总结
研究猫码中魔法与非高斯资源关系，通过比较维格纳对数负性和魔法度量分析非简并d峰猫态资源几何，发现猫码虽不继承GKP魔法-NG全局等价性，但有局部建设性对齐，为超越GKP框架的资源对应提供几何方法。
AI 中文摘要
非高斯性是连续变量量子系统中实现真正量子优势的重要资源，被视为离散变量系统中魔法资源的对应物。最近，在Gottesman-Kitaev-Preskill（GKP）编码框架内确定了非高斯性（NG）与魔法资源之间的精确对应关系。然而，这种关系在GKP编码之外是否仍然成立尚不清楚。在此，我们在猫码设置中解决这个问题。通过比较维格纳对数负性（WLN）和从相位算符基定义的魔法度量，我们分析了非简并d峰猫态的资源几何。我们发现猫码不继承全局值保持的GKP魔法-NG等价性，但它们的渐近WLN几何允许与魔法度量几何进行建设性的局部对齐。具体而言，在一个特殊的SU(d)渐近猫码下，我们基于其内在局部几何在WLN水平集和魔法度量水平集之间建立了扇区相关的对齐。这些结果确定了猫码中魔法和非高斯资源之间的全局不相容性和局部建设性关系，为超越GKP框架的资源对应提供了一种基于几何的方法。
英文摘要
Non-Gaussianity is an essential resource for genuine quantum advantages in continuous-variable quantum systems and is regarded as a counterpart of the magic resource in discrete-variable systems. Recently, an exact correspondence between non-Gaussianity (NG) and the magic resource was identified within the Gottesman--Kitaev--Preskill (GKP) encoding framework. Whether such a relation persists beyond GKP encoding, however, remains unclear. Here, we address this question in the cat-code setting. By comparing the Wigner logarithmic negativity (WLN) and a magic measure defined from a phase-operator basis, we analyze the resource geometry of non-degenerate $d$-peaked cat states. We find that cat codes do not inherit the global value-preserving GKP magic--NG equivalence, but their asymptotic WLN geometry allows a constructive local alignment with the magic-measure geometry. Specifically, under a distinguished SU($d$) asymptotic cat code, we establish a sector-dependent alignment between WLN level sets and magic-measure level sets based on their intrinsic local geometries. These results identify both the global incompatibility and the locally constructive relation between magic and non-Gaussian resources in cat codes, suggesting a geometry-based approach to resource correspondences beyond the GKP framework.
