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

TL;DR

This paper investigates how quantum control can enhance the simultaneous estimation of multiple parameters in $SU(2)$ dynamics, revealing conditions under which control improves precision and proposing strategies to optimize multiparameter quantum estimation.

Contribution

The work introduces a novel method using nested cross-products to analyze control effects on $SU(2)$ parameter estimation and identifies conditions where control enhances or loses its advantage.

Findings

Quantum control benefits depend on the angle $oldsymbol{ extalpha}_oldsymbol{ extell}$ between $oldsymbol{X}$ and $oldsymbol{ abla}_oldsymbol{ extell} oldsymbol{X}$.

Maximum control enhancement occurs when $oldsymbol{ extalpha}_oldsymbol{ extell}=rac{oldsymbol{ extpi}}{2}$.

Using entangled probes and ancillary channels can remove restrictions on simultaneous parameter estimation.

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 $\mathbf{X}$ 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 $\alpha_\ell$ between $\mathbf{X}$ and $\partial_\ell \mathbf{X}$. For an $SU(2)$ dynamics featured by $\alpha_\ell=\pi/2$, the promotion of the estimation precision can get the most benefits from the controls. When $\alpha_\ell$ gradually closes to $0$ or $\pi$, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with $\alpha_\ell=0$ or $\pi$, 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.