Minimal Tradeoff and Ultimate Precision Limit of Multiparameter 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 under the parallel scheme remains unknown. This is largely due to the lack of understandings on the incompatibility of the optimal probe states for the estimation of the three components. Here we provide an approach to characterize the minimal tradeoff among the precisions of multiple parameters that arise from the incompatibility of the optimal probe states, which leads to the identification of the ultimate precision limit for the estimation of all three components of a magnetic field under the parallel scheme. The optimal probe state that achieves the ultimate precision is also explicitly constructed. The obtained precision sets a benchmark on the precision of the multiparameter quantum magnetometry under the parallel scheme, which is of fundamental interest and importance in quantum metrology.