  An inequality about quantum noise is presented with the imprecise measurement
theory, which is used to analyse the quantum limit in continuous quantum
measurement. Different from the linear-response approach based on the quantum
relation between noise and susceptibilities of the detector, we provide an
explicit functional relation between quantum noise and reduction operator, and
show a rigorous result: The minimum noise added by the detector in quantum
measurement is precisely equal to the zero-point noise. This conclusion
generalizes the standard Haus-Caves quantum limit for a linear amplifier. We
also discuss the statistic characters of the back-action force in quantum
measurement and show on how to reach the quantum limit.
