Quantum state smoothing as an optimal Bayesian estimation problem with three different cost functions

Abstract

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.