  The precise measurement of a magnetic field is one of the most fundamental
and important tasks in quantum metrology. Although extensive studies on quantum
magnetometry have been carried out over past decades, the ultimate precision
that can be achieved for the estimation of all three components of a magnetic
field with entangled probe states under the parallel scheme remains unknown.
Here we present the ultimate lower bound for the sum of arbitrarily weighted
variances in the estimation of all three components of a magnetic field under
the parallel scheme and show that this lower bound can be achieved for
sufficiently large N. The optimal entangled probe state that achieves the
ultimate precision is also explicitly constructed. The obtained precision sets
the ultimate limit for the multi-parameter quantum magnetometry under the
parallel scheme, which is of fundamental interest and importance in quantum
metrology. Our approach also provides a way to characterize the tradeoff among
the precisions of multiple parameters that arise from the constraints on the
probe states.
