  Using standard linear response relations, we derive the quantum limit on the
sensitivity of a generic linear-response position detector, and the noise
temperature of a generic linear amplifier. Particular emphasis is placed on the
detector's effective temperature and damping effects; the former quantity
directly determines the dimensionless power gain of the detector. Unlike the
approach used in the seminal work of Caves [Phys. Rev. D, 26, 1817 (1982)], the
linear-response approach directly involves the noise properties of the
detector, and allows one to derive simple necessary and sufficient conditions
for reaching the quantum limit. Our results have direct relevance to recent
experiments on nanoelectromechanical systems, and complement recent theoretical
studies of particular mesoscopic position detectors.
