  The diverging responses to parameter variations of systems at quantum
critical points motivate schemes of quantum metrology that feature
sub-Heisenberg scaling of the sensitivity with the system size (e.g., the
number of particles). This sensitivity enhancement is fundamentally rooted in
the formation of Schr\"odinger cat states, or macroscopic superposition states
at the quantum critical points. The cat states, however, are fragile to
decoherence caused by local noises on individual particles or coupling to local
environments, since the local decoherence of any particle would cause the
collapse of the whole cat state. Therefore, it is unclear whether the
sub-Heisenberg scaling of quantum critical metrology is robust against the
local decoherence. Here we study the effects of local decoherence on the
quantum critical metrology, using a one-dimensional transverse-field Ising
model as a representative example. Based on a previous work [Phys. Rev. Lett.
94, 047201 (2005)] on the critical behaviors of the noisy Ising model, which
shows that the universality class of the quantum criticality is modified by the
decoherence, we find that the standard quantum limit is recovered by the
single-particle decoherence, which is equivalent to local quantum measurement
conducted by the environment and destroys the many-body entanglement in the
ground state at the quantum critical point. Following the renormalization group
analysis [Phys. Rev. B 69, 054426 (2004)], we argue that the noise effects on
quantum critical metrology should be universal. This works demonstrates the
importance of protecting macroscopic quantum coherence for quantum sensing
based on critical behaviors.
