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

Abstract

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 X of SU (2) generators and its partial derivative ∂ 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 between X and ∂ X. For an SU (2) dynamics featured by α = π/2, the promotion of the estimation precision can get the most benefits from the controls. When α gradually closes to 0 or π, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with α = 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.