  We provide efficient and intuitive tools for deriving bounds on achievable
precision in quantum enhanced metrology based on the geometry of quantum
channels and semi-definite programming. We show that when decoherence is taken
into account, the maximal possible quantum enhancement amounts generically to a
constant factor rather than quadratic improvement. We apply these tools to
derive bounds for models of decoherence relevant for metrological applications
including: dephasing,depolarization, spontaneous emission and photon loss.
