Quantum state estimation for continuously monitored dynamical systems involves assigning a quantum state to an individual system at some time, conditioned on the results of continuous observations. The quality of the estimation depends on how much observed information is used and on how optimality is defined for the estimate. In this work, we consider problems of quantum state estimation where some of the measurement records are not available, but where the available records come from both before (past) and after (future) the estimation time, enabling better estimates than is possible using the past information alone. Past–future information for quantum systems has been used in various ways in the literature, in particular, the quantum state smoothing, the most-likely path, and the two-state vector and related formalisms. To unify these seemingly unrelated approaches, we propose a framework for partially observed quantum systems with continuous monitoring, wherein the first two existing formalisms can be accommodated, with some generalization. The unifying framework is based on state estimation with expected cost minimization, where the cost can be defined either in the space of the unknown record or in the space of the unknown true state. Moreover, we connect all three existing approaches conceptually by defining five new cost functions, and thus new types of estimators, which bridge the gaps between them. We illustrate the applicability of our method by calculating all seven estimators we consider for the example of a driven two-level system dissipatively coupled to bosonic baths. Our theory also allows connections to classical state estimation, which create further conceptual links between our quantum state estimators.

连续监测动态系统的量子状态估计涉及在某个时间将量子状态分配给单个系统，以连续观察的结果为条件。估计的质量取决于使用了多少观察到的信息以及如何为估计定义最优性。在这项工作中，我们考虑了量子状态估计的问题，其中一些测量记录不可用，但可用记录来自估计时间之前（过去）和之后（未来），从而实现比使用过去的信息。量子系统的过去-未来信息已在文献中以各种方式使用，特别是量子态平滑、最可能的路径、二态向量和相关形式。为了统一这些看似无关的方法，我们提出了一个用于连续监测的部分观察量子系统的框架，其中可以适应前两种现有形式，并进行一些概括。统一框架基于具有预期成本最小化的状态估计，其中成本可以在未知记录的空间或未知真实状态的空间中定义。此外，我们通过定义五个新的成本函数和新类型的估计器，在概念上将所有三种现有方法联系起来，从而弥合了它们之间的差距。我们通过计算我们考虑的所有七个估计量来说明我们的方法的适用性，我们考虑了一个驱动的两级系统耗散耦合到玻色浴的例子。我们的理论还允许连接到经典状态估计，