AI 中文总结
研究基于QB主义对量子理论的解释进行逻辑形式化，通过引入相关语言和证明定理，将无动态算子片段转化为一阶公式，归约有效性，还表明有限维量子理论可通过SIC等实现框架并满足相关条件。
AI 中文摘要
QB主义将量子理论解释为关于主体概率赋值及其在可能经验中的修正的规范学科。本文对该图景进行了逻辑形式化。一个格式良好的核心数据由一个可允许的先验空间、有限的实际测量族、玻恩核和更新核组成。针对此类数据，为历史和后验状态引入了一种受保护的动态语言并证明了全局约化定理。接着考虑有效呈现的实闭域上的有效半代数数据，将无动态算子的片段转化为有序环相应语言中的一阶公式，从而将有效性归约为实闭域上的一阶推理。最后，在对称信息完备（SIC）参考测量下，表明有限维量子理论通过SIC坐标、POVM和量子仪器实现该框架，证明了相应的SIC图像满足标准qplex几何条件，且在明确系数域假设下所得量子数据是有效半代数的。
英文摘要
QBism interprets quantum theory as a normative discipline for an agent's probability assignments and their revision across possible experience. This paper develops a logical formalization of that picture. A well-formed core datum consists of an admissible prior space, a finite family of actual measurements, Born kernels, and update kernels. For each such datum, we introduce a guarded dynamic language for histories and posterior states and prove a global reduction theorem. We next consider effectively semialgebraic data over an effectively presented real closed field. For data in this class, we translate the fragment without dynamic operators into first-order formulas in the corresponding language of ordered rings, thereby reducing validity to first-order reasoning over real closed fields. Together, the reduction and first-order translation yield a sound and complete recursive calculus and a decision procedure for validity. Finally, assuming a symmetric informationally complete (SIC) reference measurement, we show that quantum theory in finite dimensions realizes the framework through SIC coordinates, POVMs, and quantum instruments. We also prove that the corresponding SIC image satisfies the standard qplex geometry conditions, namely the consistency bounds and the lower polar condition, and that under explicit coefficient field hypotheses the resulting quantum datum is effectively semialgebraic.
