Quantum state smoothing when Alice assumes the wrong type of monitoring by Bob

An open quantum system leaks information into its environment. In some circumstances it is possible for an observer, say Alice, to recover that information, as a classical measurement record, in a variety of different ways, using different experimental setups. The optimal way for Alice to estimate the quantum state at time $t$ from the record before $t$ is known as quantum filtering. Recently, a version of quantum smoothing, in which Alice estimates the state at time $t$ using her record on both sides of $t$, has been developed. It requires Alice to make optimal inferences about the pre-$t$ record of a second observer, say Bob, who recovers whatever information Alice does not. But for Alice to make this inference, she needs to know Bob's setup. In this paper we consider what happens if Alice is mistaken in her assumption about Bob's setup. We show that the accuracy -- as measured by the Trace-Squared-Deviation, of Alice's estimate of the true state (i.e., the state conditioned on her and Bob's pre-$t$ records) -- depends strongly on her setup, Bob's actual setup, and the wrongly assumed setup. Using resonance fluorescence as a model system, we show numerically that in some cases the wrong smoothing is almost as accurate as the right smoothing, but in other cases much less accurate, even being less accurate than Alice's filtered estimate. Curiously, in some of the latter cases the fidelity of Alice's wrong estimate with the true state is actually higher than that of her right estimate. We explain this, and other features we observe numerically, by some simple analytical arguments.