We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum nondemolition. Despite several claims of “measuring noncommuting observables” in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing, or retrodiction. In particular, we examine proposals to estimate the result of an external measurement made on an open quantum systems during a period where it is also undergoing continuous monitoring. A full analysis shows that the nondemolition principle is in place, and so a well-posed statistical inference problem can be formulated. We extend the formalism to consider multiple independent external measurements made on the system over the course of a continual period of monitoring.

中文翻译：

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连续监测后如何估算过去的量子测量干预

我们从数学的角度分析了估计被监视系统过去的量子态的问题，以确保与量子不爆破原理的自洽性。尽管物理学文献中多次声称“测量非通勤可观察物”，但我们表明我们一直在测量通勤过程。我们的主要兴趣在于量子平滑或倒退的概念。特别是，我们研究了在连续监测期间估算开放式量子系统上的外部测量结果的建议。全面的分析表明，不拆卸原理已经到位，因此可以提出一个恰当的统计推断问题。