  We consider charge-qubit monitoring (continuous-in-time weak measurement) by
a single-electron transistor (SET) operating in the sequential-tunneling
regime. We show that commonly used master equations for this regime are not of
the Lindblad form that is necessary and sufficient for guaranteeing valid
physical states. In this paper we derive a Lindblad-form master equation and a
corresponding quantum trajectory model for continuous measurement of the charge
qubit by a SET. Our approach requires that the SET-qubit coupling be strong
compared to the SET tunnelling rates. We present an analysis of the quality of
the qubit measurement in this model (sensitivity versus back-action).
Typically, the strong coupling when the SET island is occupied causes
back-action on the qubit beyond the quantum back-action necessary for its
sensitivity, and hence the conditioned qubit state is mixed. However, in one
strongly coupled, asymmetric regime, the SET can approach the limit of an ideal
detector with an almost pure conditioned state. We also quantify the quality of
the SET using more traditional concepts such as the measurement time and
decoherence time, which we have generalized so as to treat the strongly
responding regime.
