Prediction and retrodiction for a continuously monitored superconducting qubit

We use weak measurement to track single trajectories of a superconducting qubit embedded in a three-dimensional cavity which is subjected to continuous monitoring and driven unitary evolution. The information inferred from the measurement record is incorporated in a density matrix $\rho_{t}$, which is conditioned on probe results until $t$, and in an auxiliary matrix $E_{t}$, which is conditioned on probe results obtained after $t$. We employ a stochastic master equation to propagate $\rho_{t}$ forward in time to make predictions about weak and strong qubit measurements performed at time $t$. After these measurements, the system is subject to further probing and unitary evolution, and we propagate $E_{t}$ backward in time to make retrodictions about past measurements. Our experiments show that the predictions conditioned on $\rho$ and $E$ are more confident and nontrivially different than the predictions based only on $\rho$.