  We present a framework for the quantum enhanced estimation of multiple
parameters corresponding to non-commuting unitary generators. Our formalism
provides a recipe for the simultaneous estimation of all three components of a
magnetic field. We propose a probe state that surpasses the precision of
estimating the three components individually and discuss measurements that come
close to attaining the quantum limit. Our study also reveals that too much
quantum entanglement may be detrimental to attaining the Heisenberg scaling in
quantum metrology.
