Enhancing parameter estimation precision by non-Hermitian operator process

Abstract

Recently, Zhong et al. (Phys Rev A 87:022337, 2013) investigated the dynamics of quantum Fisher information (QFI) in the presence of decoherence. We here reform their results and propose two schemes to enhance and preserve the QFIs for a qubit system subjected to a decoherence noisy environment by applying ${non\text {-}Hermitian}$ operator process either before or after the amplitude damping noise. Resorting to the Bloch sphere representation, we derive the exact analytical expressions of the QFIs with respect to the amplitude parameter $\theta$ and the phase parameter $\phi$, and in detail investigate the influence of ${non\text {-}Hermitian}$ operator parameters on the QFIs. Compared with pure decoherence process (without non-Hermitian operator process), we find that the ${post non\text {-}Hermitian}$ operator process can potentially enhance and preserve the QFIs by choosing appropriate ${non\text {-}Hermitian}$ operator parameters, while with the help of the ${prior non\text {-}Hermitian}$ operator process one could completely eliminate the effect of decoherence to improve the parameters estimation. Finally, a generalized non-Hermitian operator parameters effect on the parameters estimation is also discussed.

Appendix A

In this appendix, we give the explicit analytical expressions of the QFIs. First we calculate the nonzero elements of ${\tilde{\rho }}_{I}(t)$

$$\begin{aligned} r_{x}= & {} \frac{2\Delta \cos ^2\alpha \cos \phi \sin \theta }{2+\Gamma \sin 2\tau \sin 2\alpha -2\sin \alpha (\cos 2\tau \sin \alpha +2\Delta \sin ^2\tau \sin \theta \sin \phi )}, \end{aligned}$$

(A1)

$$\begin{aligned} r_{y}= & {} \frac{\sin ^2\tau (2\Delta \sin \theta \sin \phi -4\sin \alpha )+\Delta \sin \theta \sin \phi (1-2\cos ^2\tau +\sin ^2\alpha )-2\Gamma \cos \alpha \sin 2\tau -\Delta \cos ^2\alpha \sin \theta \sin \phi }{2+\Gamma \sin 2\tau \sin 2\alpha -2\sin \alpha (\cos 2\tau \sin \alpha +2\Delta \sin ^2\tau \sin \theta \sin \phi )},\nonumber \\ \end{aligned}$$

(A2)

$$\begin{aligned} r_{z}= & {} \frac{2\Gamma \cos 2\tau \cos ^2\alpha +\sin 2\tau (\sin 2\alpha -2\Delta \cos \alpha \sin \theta \sin \phi )}{2+\Gamma \sin 2\tau \sin 2\alpha -2\sin \alpha (\cos 2\tau \sin \alpha +2\Delta \sin ^2\tau \sin \theta \sin \phi )}, \end{aligned}$$

(A3)

$\Gamma \equiv 1+\Delta ^2(\cos \theta -1)$. Substituting above questions into Eq. (3), the analytical expressions of the QFIs of ${\tilde{\rho }}_{I}(t)$ with respect to $\theta$ and $\phi$

$$\begin{aligned} {\mathcal {F}}_{\theta }^{}= & {} \frac{4\cos ^2\alpha }{N}\nonumber \\&\left\{ \Delta ^2\cos ^2\alpha \cos ^2\phi [\cos \theta (2-2\cos 2\tau \sin ^2\alpha )\right. \nonumber \\&+(\Delta ^2+\eta ^2\cos \theta )\sin 2\tau \sin 2\alpha ]^2+4\cos ^4 \alpha [\Delta \cos 2\tau \cos \alpha \cos \theta \sin \phi \nonumber \\&+\Delta \sin 2\tau (\Delta ^2\sin \alpha \sin \phi +\eta ^2\cos \theta \sin \alpha \nonumber \\&\sin \phi -\Delta \sin \theta )]^2+4\eta ^2\Delta ^2\cos ^2\alpha \sin ^2 \frac{\theta }{2}\nonumber \\&\times \left[ (2-\cos 2\tau +\cos 2(\tau -\alpha ))\cos \frac{\theta }{2}-4\Delta \sin ^2\tau \sin \alpha \sin \phi \sin \frac{\theta }{2}\right] ^2\nonumber \\&+\left[ \Delta \sin \phi \sin 2\alpha (\cos ^2\tau +\eta ^2\cos ^2\tau \cos \theta +\eta ^2+\eta ^2\sin ^2\tau )\right. \nonumber \\&-2\Delta \sin \phi \sin 2\tau \cos ^2\alpha \cos \theta \nonumber \\&+\left. \Delta ^2\cos \alpha (1-2\cos 2\tau -\cos 2\alpha )\sin \theta \right. \nonumber \\&\left. +\left. \Delta \sin 2\alpha \sin \phi (\Delta ^2\cos ^2\tau -\eta ^2\cos \theta (\sin ^2\tau +1)-2)\right] ^2\right\} , \end{aligned}$$

(A4)

$$\begin{aligned} {\mathcal {F}}_{\phi }^{}= & {} \frac{4\cos ^4\alpha }{N}\left\{ 256\eta ^2\Delta ^2\cos ^2\frac{\theta }{2}\cos ^2\phi \sin ^4\tau \sin ^2\alpha \sin ^6\frac{\theta }{2}\right. \nonumber \\&+16\Delta ^2\cos ^2\phi \sin ^2\tau \sin ^2\theta (\cos \tau \cos \alpha +\Gamma \sin \tau \sin \alpha )^2\nonumber \\&+4\Delta ^2\cos ^2\alpha \cos ^2\phi \sin ^2\theta (\cos 2\tau \cos \alpha +\Gamma \sin 2\tau \sin \alpha )^2\nonumber \\&+\sin ^2\theta [\Delta (2-2\cos 2\tau \sin ^2\alpha +\Gamma \sin 2\tau \sin 2\alpha \sin \phi )\nonumber \\&-\left. 4\Delta ^2\sin ^2\tau \sin \alpha \sin \theta ]^2\right\} , \end{aligned}$$

(A5)

where $N=[2-2\sin \alpha (2\Delta \sin ^2\tau \sin \theta \sin \phi +\cos 2\tau \sin \alpha )+\Gamma \sin 2\tau \sin 2\alpha ]^4$.