  Nano-mechanical resonators have a large potential as sensors of very small
adsorbed masses, down to the atomic level and beyond. Here I establish the
fundamental lower bound on the mass that can be measured with a nano-mechanical
oscillator in a given quantum state based on the quantum Cram\'er--Rao bound,
limited only by the laws of quantum mechanics, and identify the quantum states
which will allow the largest sensitivity for a given maximum energy.
