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We provide efficient and intuitive tools for deriving bounds on achievableprecision in quantum enhanced metrology based on the geometry of quantumchannels and semi-definite programming. We show that when decoherence is takeninto account, the maximal possible quantum enhancement amounts generically to aconstant factor rather than quadratic improvement. We apply these tools toderive bounds for models of decoherence relevant for metrological applicationsincluding: dephasing,depolarization, spontaneous emission and photon loss.

The elusive Heisenberg limit in quantum enhanced metrology