In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. [Phys. Rev. A 96, 062125 (2017)] the case of subsequent measurements has been treated with the derivation of the Lüders rule and its generalization (Kraus update rule). Here we investigate the special case of subsequent measurements where an intermediate measurement is a composition of two measurements ($a\text{or}b$) and the case where the causal order is not defined ($a\text{and}b\text{or}b\text{and}a$). In both cases interference effects can arise. We show that the associated probability cannot be written univocally, and the distributive property on its arguments cannot be taken for granted. The two probability expressions correspond to the Born rule and the classical probability; they are related to the intrinsic possibility of obtaining definite results for the intermediate measurement. For indefinite causal order, a causal inequality is also deduced. The frontier between the two cases is investigated in the framework of generalized measurements with a toy model, a Mach-Zehnder interferometer with a movable beam splitter.

中文翻译：

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有或没有确定因果顺序的广义测量中的条件概率和干扰

在广义测量理论的背景下，格里森-布希定理确保相关概率函数的唯一形式。最近，在Flatt等人。[物理 修订版A 96，062125（2017）]的后续测量的情况下，已经与吕德斯规则的推导及其推广（克劳斯更新规则）处理。在这里，我们调查后续测量的特殊情况，其中中间测量是两个测量的组合（$\mathrm{一种}\text{要么}b$）以及未定义因果顺序的情况（$\mathrm{一种}\text{和}b\text{要么}b\text{和}\mathrm{一种}$）。在这两种情况下，都会产生干扰效应。我们证明了关联概率不能被唯一地写，并且其论证的分配性质不能被认为是理所当然的。这两个概率表达式分别对应于伯恩法则和经典概率。它们与获得确定的中间测量结果的内在可能性有关。对于不确定的因果顺序，还推导了因果不等式。这两种情况之间的边界是在玩具模型，带有可移动分束器的Mach-Zehnder干涉仪的广义测量框架内进行研究的。