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Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED

Compared with the quantum trajectory equation, the quantum Bayesian approach has the advantage of being more efficient to infer quantum state under monitoring, based on the integrated output of measurement. For weak measurement of qubits in circuit quantum electrodynamics(cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest.Elegant work towards this task was carried out by Korotkov in "bad-cavity" and weak-response limits (arXiv:1111.4016).In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of arXiv:1111.4016 to more general system parameters.The obtained Bayesian rule is in full agreement with Korotkov's result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.