Exact quantum Bayesian rule for qubit measurements in circuit QED

Developing efficient and reliable schemes for practical quantum measurements is of essential importance to quantum information science and quantum metrology. In this work, for the increasingly important superconducting circuit-QED setup, we present a rigorous approach starting with the quantum trajectory equation (QTE) to establish an {\it exact} quantum Bayesian rule. For the „realistic“ back-action (no qubit state information gain), we obtain important correction factors for arbitrary setup parameters. For the „spooky“ information gain back-action, we establish new prior distribution knowledge for the Bayesian inference, which differ from the standard Gaussian distribution and ensure to give strictly the same results as that by numerically integrating the QTE. Compared to the QTE approach, while keeping the same accuracy, the obtained quantum Bayesian rule has much higher efficiency to compute the stochastic change of the measured state. The generic method of this work opens also a new way to construct exact quantum Bayesian rules for quantum measurement in other systems.