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Ultimate precision of multi-parameter quantum magnetometry under the parallel scheme

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.