Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called $\textit{unit separability}$, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

中文翻译：

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任何准备和测量情景的上下文的可解标准

从被称为准备和测量场景的任意一组量子状态和测量开始，构建与准备和测量场景相关的量子统计的操作上非上下文本体模型。操作上的非上下文本体模型与断层完整场景的标准 Spekkens 非上下文本体模型一致，同时用缩减空间的新概念覆盖非断层完整的情况，我们遵循非上下文的指导原则。一个名为 $\textit{unit separability}$ 的数学标准被表述为相关的经典性标准——这个名称的灵感来自于量子态可分离性的通常概念。使用这个标准，我们推导出本体空间基数的新上界。然后，我们将单元可分离性标准重新定义为一组（可能是无限的）线性约束，从中我们获得两个独立的算法测试层次结构，以见证非经典性或证明场景的经典性。最后，我们在广义概率理论的框架中重新表述了我们的结果，并讨论了这些理论中单纯形可嵌入性的含义。