  The conventional method of qubit measurements in circuit QED is employing the
dispersive regime of qubit-cavity coupling, which results in an approximated
scheme of quantum nondemolition (QND) readout. This scheme becomes problematic
in the case of strong coupling and/or strong measurement drive, owing to the
so-called Purcell effect. A recent proposal by virtue of longitudinal coupling
suggests a new scheme to realize fast, high-fidelity, and {\it ideal QND}
readout of qubit state. The aim of the present work is twofold: (i) In parallel
to what has been done in the past years for the dispersive readout, we carry
out the gradual partial-collapse theory for this recent scheme, in terms of
both the quantum trajectory equation and quantum Bayesian approaches. The
partial-collapse weak measurement theory is useful for such as the
measurement-based feedback control and other quantum applications. (ii) In the
physical aspect, we construct the joint qubit-plus-cavity entangled state under
continuous measurement and present a comprehensive analysis for the quantum
efficiency,qubit-state purity, and signal-to-noise ratio in the output
currents. The combination of the joint state and the quantum Bayesian rule
provides a generalized scheme of cavity reset associated with the longitudinal
coupling, which can restore the qubit to a quantum pure state from entanglement
with the cavity states, and thus benefits the successive partial-collapse
measurements after qubit rotations.
