PACS 85.85.+j -Micro-and nano-electromechanical systems (MEMS/NEMS) and devices PACS 07.75.+h -Mass spectrometers PACS 03.67.-a -Quantum information Abstract -I establish the fundamental lower bound on the mass that can be measured with a nano-mechanical resonator in a given quantum state based on the fundamental quantum Cramér-Rao bound, and identify the quantum states which will allow the largest sensitivity for a given maximum energy. I show that with existing carbon nanotube resonators it should be possible in principle to measure a thousandth of the mass of an electron, and future improvements might allow to reach a regime where one can measure the relativistic change of mass due to absorption of a single photon, or the creation of a chemical bond.

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