将量子模糊集扩展到密度矩阵，实现更真实的语义退相干建模。
Quantum Fuzzy Sets Revisited: Density Matrices, Decoherence, and the Q-Matrix Framework
- 用密度矩阵替代纯态，让真值覆盖整个布洛赫球
- 引入全局Q-矩阵，通过部分迹生成局部量子模糊集
- 适用于量子机器学习与量子逻辑研究者
2006年我们提出量子模糊集，指出量子寄存器状态可作为模糊子集的特征函数，将扎德的单位区间嵌入布洛赫球。该工作当时仅为初步探索。过去二十年间，该思想被应用于量子退火、直觉模糊连接词及量子机器学习，同时范畴量子力学的发展重塑了理论基础。本文重新审视该框架，提出两项主要拓展：第一，从纯态转向密度矩阵，使真值占据整个布洛赫球而非仅表面，从而刻画纯态语义无法表达的语义退相干现象；第二，引入全局Q-矩阵，个体量子模糊集作为其局部截面通过部分迹获得。我们定义了量子模糊集范畴QFS，建立基本结构性质（张量结构、在集合上的纤维化），将经典极限表征为同时对角化，并揭示完全内蕴弗罗贝尼乌斯代数处理的障碍。
原文摘要 · Abstract (English)
In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets, embedding Zadeh's unit interval into the Bloch sphere. That paper was deliberately preliminary. In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning, while parallel developments in categorical quantum mechanics have reshaped the theoretical landscape. The present paper revisits that programme and introduces two main extensions. First, we move from pure states to density matrices, so that truth values occupy the entire Bloch ball rather than its surface; this captures the phenomenon of semantic decoherence that pure-state semantics cannot express. Second, we introduce the Q-Matrix, a global density matrix from which individual quantum fuzzy sets emerge as local sections via partial trace. We define a category QFS of quantum fuzzy sets, establish basic structural properties (monoidal structure, fibration over Set), characterize the classical limit as simultaneous diagonalizability, and exhibit an obstruction to a fully internal Frobenius-algebra treatment.
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