  In quantum sensing and metrology, an important class of measurement is the
continuous linear measurement, in which the detector is coupled to the system
of interest linearly and continuously in time. One key aspect involved is the
quantum noise of the detector, arising from quantum fluctuations in the
detector input and output. It determines how fast we acquire information about
the system and also influences the system evolution in terms of measurement
backaction. We therefore often categorize it as the so-called imprecision noise
and quantum backaction noise. There is a general Heisenberg-like uncertainty
relation that constrains the magnitude of and the correlation between these two
types of quantum noise. The main result of this paper is to show that, when the
detector becomes ideal, i.e., at the quantum limit with minimum uncertainty,
not only does the uncertainty relation takes the equal sign as expected, but
also there are two new equalities. This general result is illustrated by using
the typical cavity QED setup with the system being either a qubit or a
mechanical oscillator. Particularly, the dispersive readout of a qubit state,
and the measurement of mechanical motional sideband asymmetry are considered.
