Quantum theory of interferometric length measurement with inefficient photodetectors

We find an upper bound for the sensitivity of Fabry-P\'erot interferometric length measurements using inefficient photodetectors. This bound applies to all possible quantum-mechanical states of the light entering the interferometer. It applies to standard photodetection, differenced photodetection, and homodyne detection. In the limit of large photon number we identify the optimal detection scheme and input states. They are homodyne detection with a coherent state in one input port, and a nonvacuum quadrature squeezed state in the other. No other quantum states of the input light give better sensitivity, provided that the interferometer output is measured by either photodetection or homodyning, and that the photodetectors are inefficient.