This paper derives bounds on the asymptotic expected distortion (high rate) for quantization on the complex projective space denoted as CPn-1. In essence the problem of quantization in an Euclidean space with constraints can be posed as an unconstrained problem on an appropriate manifold. CPn-1 is a non-linear manifold that represents the constraints that arise in areas such as communication with multiple antennas at the transmitter and receiver. Due to the constraints, the distortion analysis developed for Euclidean spaces cannot be applied directly. The special structure of CPn-1 and the distortion measures that are defined on it differentiate this problem from traditional vector quantization in Euclidean spaces.