Prediction and retrodiction for a continuously monitored superconducting   qubit
  The quantum state of a superconducting transmon qubit inside a three-dimensional cavity is monitored by reflection of a microwave field on the cavity. 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 $t$. Here, we obtain these matrices from experimental data and we illustrate their application to predict and retrodict the outcome of weak and strong qubit measurements. 
D. Tan, S. Weber, I. Siddiqi, K. M{\o}lmer, K. W. Murch
quant-ph cond-mat.mes-hall cond-mat.supr-con