Quantum state smoothing is a technique to construct an estimate of the quantum state at a particular time, conditioned on a measurement record from both before and after that time. The technique assumes that an observer, Alice, monitors part of the environment of a quantum system and that the remaining part of the environment, unobserved by Alice, is measured by a secondary observer, Bob, who may have a choice in how he monitors it. The effect of Bobs measurement choice on the effectiveness of Alices smoothing has been studied in a number of recent papers. Here we expand upon the Letter which introduced linear Gaussian quantum (LGQ) state smoothing [Phys. Rev. Lett., 122, 190402 (2019)]. In the current paper we provide a more detailed derivation of the LGQ smoothing equations and address an open question about Bobs optimal measurement strategy. Specifically, we develop a simple hypothesis that allows one to approximate the optimal measurement choice for Bob given Alices measurement choice. By optimal choice we mean the choice for Bob that will maximize the purity improvement of Alices smoothed state compared to her filtered state (an estimated state based only on Alices past measurement record). The hypothesis, that Bob should choose his measurement so that he observes the back-action on the system from Alices measurement, seems contrary to ones intuition about quantum state smoothing. Nevertheless we show that it works even beyond a linear Gaussian setting.
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