Enhancing the precision of parameter estimation in a dissipative qutrit via quantum feedback control and classical driving

Abstract

In this paper, we use quantum feedback control and classical driving to enhance the precision of parameter estimation in a dissipative qutrit by investigating its dynamics of quantum Fisher information. We investigate the parameter estimation of a dissipative qutrit in a cavity which provides the quantum feedback to the qutrit. The results show that the precision of the parameter to be estimated can be effectively enhanced via the local quantum feedback control based on quantum-jump detection. Particularly, the higher parameter estimation accuracy can be achieved when the feedback control and classical driving are present at the same time.

Appendices

A Appendix A: quantum Fisher information

In this section, we will review the basic properties of QFI. We assume $\varphi$ to be a single parameter and $P_{i}(\varphi )$ to be the probability density with measurement outcome {$x_{i}$} for a discrete observable X conditioned on the fixed parameter $\varphi$. The bound of the variance Var(${\hat{\varphi }}$) for an unbiased estimator ${\hat{\varphi }}$ is given by the classical Cramer–Rao inequality [6]. The classical Cramer–Rao inequality gives the bound of the variance Var(${\hat{\varphi }}$) for an unbiased estimator ${\hat{\varphi }}$,

$$\begin{aligned} Var({\hat{\varphi }}) \ge \frac{1}{H_{\varphi }} \end{aligned}$$

(11)

where the classical Fisher information is given as [40] $H_{\varphi }=\sum _{i}P_{i}(\varphi )[\frac{\partial }{\partial \varphi }lnP_{i}(\varphi )]^2$.

The symmetric logarithmic derivative $L_{\varphi }$ is determined by $\frac{\partial \rho _{\varphi }}{\partial \varphi }=\frac{1}{2}(\rho _{\varphi }L_{\varphi }+L_{\varphi }\rho _{\varphi })$ in order to extend to quantum regime. Consequently, a bound of the variance of unbiased estimates is given by the so-called quantum Cramer–Rao inequality [41]:

$$\begin{aligned} Var({\hat{\varphi }}) \ge \frac{1}{H_{\varphi }} \ge \frac{1}{F_{\varphi }} \end{aligned}$$

(12)

What is worth mentioning is that an optimal metric is provided by symmetric logarithmic derivative $L_{\varphi }$. Here, the QFI of the parameter $\varphi$ can be given as [42]

$$\begin{aligned} F_{\varphi }=Tr[\rho _{\varphi }L_{\varphi }^2] \end{aligned}$$

(13)

In addition, taking advantage of the spectrum decomposition $\rho _{\varphi }=\sum _{k} \lambda _{k}|k\rangle \langle k|$, the QFI can be represented as

$$\begin{aligned} F_{\varphi }= & {} \sum _{k,\lambda _{k} \ge 0}\frac{(\partial _{\varphi }\lambda _{k})^2}{\lambda _{k}}+ \sum _{k,k^{'},\lambda _{k}+\lambda _{k^{'}} \ge 0} \frac{2(\lambda _{k}-\lambda _{k^{'}})^2}{\lambda _{k}+\lambda _{k^{'}}}|\langle k|\partial _{\varphi }k^{'}\rangle |^2. \end{aligned}$$

(14)

The first term can be considered as the classical Fisher information, and the second term in this equation is just the quantum contribution. And the expression of the symmetric logarithmic derivative can be written as

$$\begin{aligned} L_{\varphi }=\sum _{k,k^{'},\lambda _{k}+\lambda _{k^{'}} \ge 0} \frac{2\langle k|\partial _{\varphi }k^{'}\rangle }{\lambda _{k}+\lambda _{k^{'}}}|k\rangle \langle k^{'}|. \end{aligned}$$

(15)

Moreover, the QFI is also related to the Bures distance, which can be expressed as [41]

$$\begin{aligned} D_{B}^2[\rho _{\varphi },\rho _{\varphi +d\varphi }]=\frac{1}{4}F_{\varphi }d\varphi ^2, \end{aligned}$$

(16)

Here, $D_{B}^2[\rho ,\sigma ]=[2(1-Tr\root \of {\rho ^{1/2}\sigma \rho ^{1/2}})]^{1/2}$ defines the Bures distance, which measures the distance between two quantum states $\rho$ and $\sigma$ .

B Appendix B: Solution of Eq. (9)

Without driving, the evolved density matrix of the qutrit can be exactly solved, which can be given as

$$\begin{aligned} \rho _{11}(t)= & {} \exp {(-t\gamma _{1}+t\alpha _{1})} \nonumber \\ \rho _{12}(t)= & {} i\left[ \exp {(-\frac{1}{2}t \gamma _{1}-\frac{1}{2}t\gamma _{2})}-\exp {(t\alpha _{1}-t\gamma _{1} )} \right] \eta _{1} \nonumber \\&\qquad \times&\gamma _{1}(\gamma _{2}-\gamma _{1}+2\alpha _{1})^{-1} \sin (2\lambda _{23}) \nonumber \\ \rho _{13}(t)= & {} 0 \nonumber \\ \rho _{22}(t)= & {} \left[ \exp {(-t\gamma _{1}+t\alpha _{1})} - \exp {(-t\gamma _{2}+t\alpha _{2})} \right] (\gamma _{1}-\alpha _{1}) \nonumber \\&\qquad \times&(\beta _{1}-\beta _{2})^{-1} \nonumber \\ \rho _{23}(t)= & {} i\left[ 2\exp {(\frac{1}{2}t\gamma _{2})(\beta _{1}{-}\beta _{2})}{+}\exp {(t\alpha _{1}-t\gamma _{1}{+}t\gamma _{2})}(\alpha _{2}{+}\beta _{2}){-}\exp {(t\alpha _{2})} (2\beta _{1}{+}\gamma _{2})\right] \nonumber \\&\qquad \times&\eta _{2} \beta _{1} \exp {(-t\gamma _{2})} \sin (2\lambda _{12}) \left[ (\beta _{1}-\beta _{2})(2\beta _{1}+\gamma _{2})(2\eta _{2}\sin (\lambda _{12})^2-1) \right] ^{-1} \nonumber \\ \rho _{33}(t)= & {} 1-\rho _{22}(t)-\rho _{11}(t) \quad \rho _{21}(t)=\rho _{12}(t)^{*} \nonumber \\ \rho _{32}(t)= & {} \rho _{23}(t)^{*} \quad \rho _{31}(t)=\rho _{13}(t)^{*} \end{aligned}$$

where $\alpha _{1}= \gamma _{1} \eta _{1} \sin (\lambda _{23})^2$, $\alpha _{2}= \gamma _{2} \eta _{2} \sin (\lambda _{12})^2$, $\beta _{1}=\alpha _{1}-\gamma _{1}$, and $\beta _{2}=\alpha _{2}-\gamma _{2}$. When $\eta _{1}=\eta _{2}=1$, the above equation becomes Eq. (7).

With driving, we cannot solve the time-evolving density matrix of the qutrit analytically. However, one can obtain the density matrix elements of the steady state in the long-time limit as

$$\begin{aligned} \rho _{11}= & {} 16\varOmega _{1}^2\varOmega _{2}^2(\varepsilon _{6}-\eta _{2}\gamma _{2}+\varepsilon _{1})/N \nonumber \\ \rho _{12}= & {} -8i\varOmega _{1}\varOmega _{2}^2(\gamma _{1}-\eta _{1}\gamma _{1}+\varepsilon _{2})(\varepsilon _{6}-\eta _{2}\gamma _{2})/N \nonumber \\ \rho _{13}= & {} -4\varOmega _{1}\varOmega _{2}\left[ 1/2\varepsilon _{2}\varepsilon _{6}(\varepsilon _{3}+2\gamma _{2}-\eta _{2}\gamma _{2})- (\varepsilon _{1}-\varepsilon _{9})(\varepsilon _{6}\varepsilon _{8}-4\varOmega _{1}^2)\right. \nonumber \\ \rho _{22}= & {} 2\gamma _{1}\varOmega _{2}^{2}\left[ 8\varepsilon _{7}{-}(\eta _{1}-2)\varepsilon _{6}\gamma _{1}{-}4(\varepsilon _{4}\varOmega _{1}+\eta _{1}\varOmega _{2}^2) +\eta _{1}\cos (2\lambda _{23})(\varepsilon _{6}\gamma _{1}+4\varOmega _{2}^2) \right] /N \nonumber \\ \rho _{23}= & {} -i\left[ 8\varepsilon _{7}-(\eta _{1}-2)\varepsilon _{6}\gamma _{1}-4(\varepsilon _{4}\varOmega _{1}+\eta _{1}\varOmega _{2}^2) +\eta _{1}\cos (2\lambda _{23})(\varepsilon _{6}\gamma _{1}+4\varOmega _{2}^2) \right] \nonumber \\&\qquad \times&\gamma _{1}\gamma _{2}\varOmega _{2}(1-\eta _{2}\sin (\lambda _{12})^2)/N \nonumber \\ \rho _{33}= & {} 1-\rho _{22}-\rho _{11} \quad \rho _{21}=\rho _{12}^{*} \nonumber \\ \rho _{32}= & {} \rho _{23}^{*} \qquad \rho _{31}=\rho _{13}^{*} \end{aligned}$$

(17)

with $\varepsilon _{1}=\eta _{2}\gamma _{2}\cos (\lambda _{12})^2$, $\varepsilon _{2}=\eta _{1}\gamma _{1}\cos (\lambda _{23})^2$, $\varepsilon _{3}=\eta _{2}\gamma _{2}\cos (2\lambda _{12})$, $\varepsilon _{4}=\eta _{1}\gamma _{1}\sin (2\lambda _{23})$, $\varepsilon _{5}=\eta _{2}\gamma _{2}\sin (2\lambda _{12})$, $\varepsilon _{6}=\gamma _{1}+\gamma _{2}$, $\varepsilon _{7}=\varOmega _{1}^2+\varOmega _{2}^2$, $\varepsilon _{8}=(\eta _{1}-1)\gamma _{1}$, $\varepsilon _{9}=(\eta _{2}-1)\gamma _{2}$, $\varepsilon _{10}=\gamma _{1}\gamma _{2}+4\varOmega _{1}^2$, $\nu _{1}=2(\eta _{2}-1)\varepsilon _{8}-(2-\eta _{2})\varepsilon _{2}$, $\nu _{2}=\varepsilon _{3}-(\eta _{2}-2)\gamma _{2}$, $\nu _{3}=\gamma _{1}(\gamma _{2}-\eta _{1}\gamma _{2})+4\varOmega _{1}^2$, $\nu _{4}=2\gamma _{1}^2+2\gamma _{1}\gamma _{2}+\gamma _{2}^2-\eta _{2}\gamma _{2}^2$, $\nu _{5}=8\varOmega _{1}^2-4\varepsilon _{6}\gamma _{1}+(\eta _{2}-2)\gamma _{2}^2$, $\nu _{6}=\varepsilon _{9}\varOmega _{1}+\gamma _{1}(\varepsilon _{4}-2\varOmega _{1}+\eta _{1}\varOmega _{1})$ and $N=\frac{1}{2}\left[ \varepsilon _{10}\varepsilon _{6}\nu _{1}+\varepsilon _{2}\varepsilon _{3}(4\varepsilon _{7}+\varepsilon _{6}\gamma _{1}) \right] \gamma _{2}-\varepsilon _{10}\varepsilon _{4}\nu _{2}\varOmega _{1}+2(\varepsilon _{2}\varepsilon _{3}\gamma _{1}-2\varepsilon _{10}\varepsilon _{9})\varOmega _{1}^2 +\varepsilon _{1}\varepsilon _{10}(4\varOmega _{1}^2-\varepsilon _{6}\varepsilon _{8})+2\varepsilon _{5}\gamma _{1}(2\varepsilon _{4}\varOmega _{1}-\varepsilon _{2}\varepsilon _{6}-4\varepsilon _{7}+\varepsilon _{6}\varepsilon _{8})\varOmega _{2} +4\nu _{3}\varepsilon _{1}\varOmega _{2}^2-2\varOmega _{2}^2(2\varepsilon _{8}\nu _{4}+\varepsilon _{2}\nu _{5}+8\varOmega _{1}\nu _{6})-8\varepsilon _{5}\varOmega _{2}^3(\varepsilon _{2}-\eta _{1}\gamma _{1})+32\varOmega _{2}(\varepsilon _{2}-\varepsilon _{8})$. When $\eta _{1}=\eta _{2}=1$, the above equation becomes Eq. (10).