Sensory hair cells are known for the exquisite displacement sensitivity with which they detect the sound-evoked vibrations in the inner ear. In this article, we determine a stochastically imposed fundamental lower bound on a hair cell's sensitivity to detect mechanically coded information arriving at its hair bundle. Based on measurements of transducer current and its noise in outer hair cells and the application of estimation theory, we show that a hair cell's transducer current carries information that allows the detection of vibrational amplitudes with an accuracy on the order of nanometers. We identify the transducer channel's molecular gating force as the physical factor controlling this accuracy in proportion to the inverse of its magnitude. Further, we show that the match of stochastic channel noise to gating-spring noise implies that the gating apparatus operates at the threshold of negative stiffness.