  In recent years, many peculiar sensors have been proposed, such as the
sensors based on exceptional points, Parity-Time symmetric structures or
non-reciprocal systems. It is crucial to evaluate the fundamental limit of
these sensing schemes and to judge whether there is an enhanced performance.
Several papers have already investigated the fundamental limits based on
different aspects and criteria, some led to different conclusions. In this
paper, we suggest that for linear bosonic sensors that can be modeled as
coupled oscillators below threshold, a measurement of the mode frequency $w_0$
can not has a precision beyond $\frac{\sqrt{\kappa_0}}{2\sqrt{n\tau}}$, in
which $\kappa_{0}$ is the intrinsic loss, $n$ is the average particle number at
that mode and $\tau$ is the measurement time. Such a precision limit can
already be achieved for a single mode passive sensor, and we have proved that
it can not be exceeded by adding gain components, or by coupling it to other
modes to form a n-mode linear sensor. Further more, we recognize that the above
limit is comparative to the frequency uncertainty
($\frac{\sqrt{\kappa_0}}{\sqrt{2 n\tau}}$) for an above-threshold oscillation
mode (e.g. a laser cavity above threshold), which is related to the
Schawlow-Townes laser linewidth limit. This observation makes us conjecture
that the limit we proposed for linear sensors may be a basic general limit
which restricts most, if not all, sensors.
