Quantum state smoothing is a technique to estimate an unknown true state of an open quantum system based on partial measurement information both prior and posterior to the time of interest. In this paper, we show that the smoothed quantum state is an optimal Bayesian state estimator, that is, it minimizes a Bayesian expected cost function. Specifically, we show that the smoothed quantum state is optimal with respect to two cost functions: the trace-square deviation from and the relative entropy to the unknown true state. However, when we consider a related cost function, the linear infidelity, we find, contrary to what one might expect, that the smoothed state is not optimal. For this case, we derive the optimal state estimator, which we call the lustrated smoothed state. It is a pure state, the eigenstate of the smoothed quantum state with the largest eigenvalue. We illustrate these estimates with a simple system, the driven, damped two-level atom.

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量子状态平滑作为具有三个不同成本函数的最优贝叶斯估计问题

量子状态平滑是一种基于感兴趣时间之前和之后的部分测量信息来估计开放量子系统的未知真实状态的技术。在本文中，我们展示了平滑量子态是一个最优贝叶斯状态估计器，也就是说，它最小化了贝叶斯期望成本函数。具体来说，我们表明平滑的量子状态对于两个成本函数是最佳的：来自未知真实状态的迹平方偏差和相对熵。然而，当我们考虑相关的成本函数时，线性不保真，我们发现，与人们可能预期的相反，平滑状态不是最佳的。对于这种情况，我们推导出最优状态估计器，我们称之为 lustrated 平滑状态。它是一种纯粹的状态，具有最大特征值的平滑量子态的特征态。我们用一个简单的系统来说明这些估计，驱动的、阻尼的两级原子。