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Multiparameter simultaneous optimal estimation with an SU(2) coding unitary evolution

In a ubiquitous $SU(2)$ dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this $SU(2)$ coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross-products of the coefficient vector $\mathbf{X}$ of $SU(2)$ generators and its partial derivative $\partial_\ell \mathbf{X}$, to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an $SU(2)$ dynamics with respect to the objective parameter. This characterization is quantified by the angle $α_\ell$ between $\mathbf{X}$ and $\partial_\ell \mathbf{X}$. For an $SU(2)$ dynamics featured by $α_\ell=π/2$, the promotion of the estimation precision can get the most benefits from the controls. When $α_\ell$ gradually closes to $0$ or $π$, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with $α_\ell=0$ or $π$, quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-$1/2$ system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary $SU(2)$ parametrization process.