Evaluating the quantum Ziv&#x2013;Zakai bound for phase estimation in noisy environments

Shoukang Chang; Wei Ye; Xuan Rao; Huan Zhang; Liqing Huang; Mengmeng Luo; Yuetao Chen; Shaoyan Gao; Shaoyan Gao; Liyun Hu; Liyun Hu

doi:10.1364/OE.459659

1. Introduction

Fundamental principles of quantum mechanics, e.g., Heisenberg uncertainty, impose ultimate precision limits on the parameter estimation [1–4]. To efficiently quantify the minimum estimation error in quantum metrology, the quantum Cramér-Rao bound (QCRB) is particularly famous for giving a method to derive the asymptotically attainable estimation precision [1,5–7]. More specifically, the QCRB is inversely proportional to the quantum Fisher information (QFI), so that such a lower bound plays more important roles in metrologic applications, such as quantum sensing [8–10], gravitational wave detection [11,12] and optical imaging [13–15]. Especially, with the help of the multiparameter QCRB corresponding to the QFI matrix, T.J. Proctor et al. investigated the multiparameter estimation in the framework of networked quantum sensors [16]. However, the QCRB is asymptotically tight only under the limit of infinitely many trials, which may seriously underestimate the estimation precision if the likelihood function is highly non-Gaussian [17–21]. Thus, it is still an open problem to solve the bound of evaluation accuracy for the limited number of tests or non-Gaussian cases.

For this reason, the quantum Weiss-Weinstein bound [18] and the QZZB [17,19] are often viewed as an alternative candidate to solve the problem mentioned above. Compared to the former, the later has been widely studied because it is relatively easy to calculate. For the single-parameter estimation, V. Giovannetti et al. demonstrated how to obtain a lower bound from prior information regimes, indicating that the sub-Heisenberg estimation strategies are ineffective [19]. After that, by relying on the QZZB, Y. Gao and H. Lee theoretically derived the generalized limits for parameter sensitivity when considering the implementations of adaptive measurements, and they found that the precision of phase estimation with several known states cannot be superior to the Heisenberg limit [21]. By extending the QZZB into the multiparameter cases, Y. R. Zhang and H. Fan presented two kinds of metrological lower bounds using two different approximations proposed by Tsang even in noisy systems [20]. They showed the advantage of simultaneous estimation over optimal individual one, but the achievable bounds may be not tight either due to some approximation methods used. Additionally, in order to further develop the QZZB, D. W. Berry et al. proposed a novel lower bound for phase waveform estimation as an application of multiparameter case via the quantum Bell-Ziv–Zakai bound [22]. These results show that the QZZB become one of the promising candidates to attain the lower bound of the (multi-)parameter estimation for tightness.

On the other hand, for realistic scenarios, because of the inevitable interactions between the quantum system and its surrounding noisy environment, the corresponding estimation precision would be reduced, which has been studied extensively in recent years [23–31]. In particular, since a variational method was first proposed by Escher [24], the analytical QCRB of single-(or multiple-) parameter estimation in noisy environment can be derived effectively [32–34]. We also noticed that the effects of noisy environment on the performance of the QZZB for phase estimation has not been studied before. Thus, in this paper, we shall focus on the general derivation of the QZZB for phase estimation in the presence of both the photon loss and the phase diffusion with the help of the variational method and the technique of integration within an ordered product of operators (IWOP) [35–41]. It is noteworthy that the QZZB in noisy environments is not fully guaranteed to be tight, but can be as tight as possible by optimizing the value of the variational parameter with the assistance of the variational method [24,32]. Further, in this asymptotically tight situation, we also present the estimation performance of the QZZB in the presence of the two noise scenarios when given some optical resources, such as a coherent state (CS), a single-mode squeezed vacuum state (SMSVS) and a two-mode squeezed vacuum state (TMSVS). The results show that for the photon-loss scenario, the CS shows the better estimation performance of the QZZB, due to its robustness against the photon losses, when comparing to the cases of other optical resources. While for the phase-diffusion scenario, the estimation performance of the QZZB for the CS can be outperformed by that for the TMSVS at the large range of the phase-diffusion strength.

This paper is arranged as follows. In section 2, we briefly review the known results of the QZZB. Based on the variational method and the IWOP technique, in sections 3 and 4, we respectively derive the asymptotically tight QZZB for phase estimation in the presence of the photon-loss and phase-diffusion scenarios, and then also investigate the phase estimation performance of the QZZB with the two noise scenarios for given some Gaussian states, such as the CS, the SMSVS and the TMSVS. Finally, the main conclusions are drawn in the last section.

2. QZZB

It is pointed out that the QZZB can show much tighter than the conventional QCRB for the highly non-Gaussian regime [17–19]. So, in this section, we briefly review the known results of the quantum parameter estimation based on the QZZB. From the perspective of a classical parameter-estimated theory, let $x$ be the unknown parameter to be estimated, $y$ be the observation with finite measurements, and $X\left ( y\right )$ be an estimator of $x$ constructed from the observation $y$. Thus, the parameter sensitivity of $x$ can be quantified using the mean-square estimation error

(1)$$\sum =\int dxdyp\left( y|x\right) p(x)[X\left( y\right) -x]^{2},$$

where $p\left ( y|x\right )$ represents the condition probability density of achieving the observation $y$ given $x$, and $p(x)$ represents the prior probability density. According to Refs. [42–45], a classical Ziv-Zakai bound, i.e., a lower bound for $\sum$, can be given by

(2)$$\sum \geq \int_{0}^{\infty }d\beta \frac{\beta }{2}\chi \int_{-\infty }^{\infty }dx2\min [p(x),p(x+\beta )]\Pr\nolimits_{e}^{el}(x,x+\beta ),$$

where $\Pr \nolimits _{e}^{el}(x,x+\beta )$ represents the minimum error probability for the binary decision problem with an equally prior probability, and $\chi$ is the so-called optional valley-filling operation, i.e., $\chi f(\beta )\equiv \max _{\eta \geq 0}f(\beta +\eta )$, which renders the bound tighter but harder to calculate [17,42,43].

Likewise, for the quantum parameter-estimated theory, let $\hat {\rho }\left ( x\right )$ be the density operator as a function of the unknown parameter $x$, and let $\hat {E}(y)$ be the positive operator-valued measure so as to establish the measurement model. Then, the observation density is denoted as

(3)$$p\left( y|x\right) =\text{Tr}[\hat{\rho}\left( x\right) \hat{E}(y)],$$

with the symbol of Tr being the operator trace. Thus, according to Refs. [17,20,46], a lower bound of the minimum error probability $\Pr \nolimits _{e}^{el}(x,x+\beta )$ is given by

(4)$$\begin{aligned} \Pr \nolimits_{e}^{el}(x,x+\beta ) &\geq \frac{1}{2}(1-\frac{1}{2}\left \Vert \hat{\rho}\left( x\right) -\hat{\rho}\left( x+\beta \right) \right \Vert _{1})\\ &\geq \frac{1}{2}[1-\sqrt{1-F(\hat{\rho}\left( x\right) ,\hat{\rho}\left( x+\beta \right) )}], \end{aligned}$$

with the trace norm $\left \Vert \hat {O}\right \Vert _{1}=$Tr$\sqrt {\hat {O} ^{\dagger}\hat {O}}$ and the Uhlmann fidelity given by $F(\hat {\rho }\left ( x\right ) ,\hat {\rho }\left ( x+\beta \right ) )=(Tr\sqrt {\sqrt {\hat {\rho }\left ( x\right ) }\hat {\rho }\left ( x+\beta \right ) \sqrt {\hat {\rho }\left ( x\right ) }} )^{2}$.

Now, let us assume that an unknown parameter $x$ is encoded into the quantum state $\hat {\rho }\left ( x\right )$, which can be presented in terms of an unitary evolution

(5)$$\hat{\rho}\left( x\right) =e^{{-}i\hat{H}x}\hat{\rho}e^{i\hat{H}x},$$

where $\hat {\rho }$ is the initial state and $\hat {H}$ is the effective Hamiltonian operator. Thus, it can be seen that $F(\hat {\rho }\left ( x\right ) ,\hat {\rho }\left ( x+\beta \right ) )\geq \left \vert \text {Tr}(\hat {\rho }e^{-i \hat {H}\beta })\right \vert ^{2}$ [47]$.$ After assuming that the prior probability density $p(x)$ is the uniform window with the mean $\mu$ and the width $W$ denoted as

(6)$$p(x)=\frac{1}{W}rect\left( \frac{x-\mu }{W}\right) ,$$

this implies that we have no prior information about the unknown parameter preceding the estimation [20]. In fact, this assumption is rational since it has been proved that at the condition of high prior information, the resulting precision is the same order as that obtained by this assumption [19–21].

Then omitting the optional $\chi$, one can obtain the QZZB [17], i.e.,

(7)$$\sum \geq \sum\nolimits_{Z}=\int_{0}^{W}d\beta \frac{\beta }{2}\left( 1- \frac{\beta }{W}\right) [1-\sqrt{1-F(\beta )}],$$

where $F(\beta )\equiv \left \vert \text {Tr}(\hat {\rho }e^{-i\hat {H}\beta })\right \vert ^{2}$ is the lower bound of the fidelity $F(\hat {\rho }\left ( x\right ) ,\hat {\rho }\left ( x+\beta \right ) ).$ For the initial pure state $\hat {\rho },$ $F(\beta )$ is the fidelity between $\hat {\rho }\left ( x\right )$ and $\hat {\rho }\left ( x+\beta \right ) .$ In the following, we would take the effects of the noisy environment on the QZZB for phase estimation into account, since the encoding process of the quantum state $\hat {\rho }$ to an unknown phase $\phi$ is inevitably affected by its surrounding environment, such as the photon loss and the phase diffusion.

3. Effects of photon losses on the QZZB for phase estimation

In the case of photon losses, the encoding process of the quantum state to an unknown phase $\phi$ no longer satisfies unitary evolution, so that the QZZB can not be directly derived by using Eq. (7). Fortunately, similar to obtain the upper bound for QFI proposed by Escher with the assistance of an variational method [24,48], combining with the IWOP technique, here we shall present the derivation of the QZZB in the presence of photon losses.

In order to change the encoding process into the unitary evolution $\hat {U} _{S+E}\left ( \phi \right )$, the basic idea is to introduce additional degrees of freedom, acting as an environment $E$ for the system $S$. When given an initial pure state $\hat {\rho }_{S}=\left \vert \psi _{S}\right \rangle \left \langle \psi _{S}\right \vert$ of a probe system $S,$ the encoding process of the initial pure state $\hat {\rho }_{S}$ is the non-unitary evolution under the photon losses. So, it is necessary to expand the size of the Hilbert space $S$ together with the photon-loss environment space $E$. After the quantum state in the enlarged space $S+E$ goes through the unitary evolution $\hat {U}_{S+E}\left ( \phi \right ) ,$ one can obtain

(8)$$\begin{aligned} \hat{\rho}_{S+E}\left( \phi \right) &=\left\vert \psi _{S+E}\left( \phi \right) \right\rangle \left\langle \psi _{S+E}\left( \phi \right) \right\vert\\ &=\hat{U}_{S+E}\left( \phi \right) \hat{\rho}_{S}\otimes \hat{\rho}_{E_{0}} \hat{U}_{S+E}^{{\dagger} }\left( \phi \right)\\ &=\sum_{l=0}^{\infty }\hat{\Pi}_{l}\left( \phi \right) \hat{\rho} _{S}\otimes \hat{\rho}_{E_{l}}\hat{\Pi}_{l}^{{\dagger} }\left( \phi \right) , \end{aligned}$$

where $\hat {\rho }_{E_{0}}=\left \vert 0_{E}\right \rangle \left \langle 0_{E}\right \vert$ is the initial state of the photon-loss environment space $E$, $\hat {\rho }_{E_{l}}=\left \vert l_{E}\right \rangle \left \langle l_{E}\right \vert$ is the orthogonal basis of the $\hat {\rho }_{E_{0}},$ and $\hat {\Pi }_{l}\left ( \phi \right )$ is the Kraus operator acting on the $\hat { \rho }_{S},$ which can be described as

(9)$$\hat{\Pi}_{l}\left( \phi \right) =\sqrt{\frac{\left( 1-\eta \right) ^{l}}{l!} }e^{{-}i\phi \left( \hat{n}-\lambda _{1}l\right) }\eta ^{\frac{\hat{n}}{2}} \hat{a}^{^{l}},$$

with the strength of photon losses $\eta$, the variational parameter $\lambda _{1},$ and the photon number operator $\hat {n}=\hat {a}^{\dagger}\hat {a}$. Note that $\eta =0$ and $\eta =1$ respectively correspond to complete absorption and non-loss case. According to the Uhlmann’s theorem [49,50] , thus, the fidelity for the enlarged system $S+E$ with photon losses can be given by

(10)$$F_{L_{1}}(\beta )=\max_{\left\{ \left\vert \psi _{S+E}\left( \phi \right) \right\rangle ,\left\vert \psi _{S+E}\left( \phi +\beta \right) \right\rangle \right\} }F_{Q_{1}}(\hat{\rho}_{S+E}\left( \phi \right) ,\hat{ \rho}_{S+E}\left( \phi +\beta \right) ),$$

where the maximization runs over all purifications $\left \vert \psi _{S+E}\left ( \phi \right ) \right \rangle$ of $\hat {\rho }_{S+E}\left ( \phi \right )$ and $\left \vert \psi _{S+E}\left ( \phi +\beta \right ) \right \rangle$ of $\hat {\rho }_{S+E}\left ( \phi +\beta \right )$ in the enlarged space $S+E,$ and

(11)$$\begin{aligned} &F_{Q_{1}}(\hat{\rho}_{S+E}\left( \phi \right) ,\hat{\rho}_{S+E}\left( \phi +\beta \right) )\\ = &\left\vert \left\langle \psi _{S+E}\left( \phi \right) |\psi _{S+E}\left( \phi +\beta \right) \right\rangle \right\vert ^{2}\\ = &\left\vert \text{Tr}(\hat{\rho}_{S}\hat{Z})\right\vert ^{2}, \end{aligned}$$

corresponds to the lower bound of the fidelity in photon losses with $\hat {Z}$ defined as

(12)$$\hat{Z}=\sum_{l=0}^{\infty }\hat{\Pi}_{l}^{{\dagger} }\left( \phi \right) \hat{ \Pi}_{l}\left( \phi +\beta \right) .$$

By invoking the IWOP technique and Eqs. (9) and (12), one can respectively obtain the operator identities, i.e.,

(13)$$\eta ^{\hat{n}}e^{{-}i\beta \hat{n}} =\; \colon \;\exp \left[ \left( \eta e^{{-}i\beta }-1\right) \hat{n}\right] \colon ,$$

(14)$$\hat{Z}=\left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] ^{\hat{n}},$$

where $:\cdot :$ denotes the symbol of the normal ordering form. The more details for the derivation can be seen in the Appendix A. Thus, based on Eqs. (13) and (14), the lower bound of the fidelity, i.e., $F_{Q_{1}}(\hat {\rho }_{S+E}\left ( \phi \right ) ,\hat {\rho }_{S+E}\left ( \phi +\beta \right ) )\equiv F_{Q_{1}}\left ( \beta \right ) ,$ can be given by

(15)$$F_{Q_{1}}\left( \beta \right) =\left\vert \left\langle \psi _{S}\right\vert \left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] ^{\hat{n}}\left\vert \psi _{S}\right\rangle \right\vert ^{2}.$$

From Eq. (15), it should be noted that, when the variational parameter $\lambda _{1}$ takes the optimal value $\lambda _{1opt}$ to make the lower bound of the fidelity as tight as possible, $F_{Q_{1}}\left ( \beta \right )$ can reach the maximum value, which is the fidelity (denoted as $F_{L_{1}}(\beta )$) in the photon-loss environment. In this situation, the lower limit of the minimum error probability $\Pr \nolimits _{e_{L_{1}}}^{el}( \phi,\phi +\beta )$ under the photon losses can be expressed as

(16)$$\begin{aligned} \Pr\nolimits_{e_{L_{1}}}^{el}(\phi ,\phi +\beta ) &\geq \frac{1}{2}(1-\frac{ 1}{2}\left\Vert \rho _{S+E}\left( \phi \right) +\rho _{S+E}\left( \phi +\beta \right) \right\Vert _{1})\\ &\geq \frac{1}{2}[1-\sqrt{1-F_{L_{1}}(\beta )}]. \end{aligned}$$

According to Eq. (6), finally, we can obtain the QZZB for phase estimation in the presence of the photon-loss environment

(17)$$\sum\nolimits_{L_{1}}\geq \sum\nolimits_{Z_{L_{1}}}=\int_{0}^{W}d\beta \frac{ \beta }{2}\left( 1-\frac{\beta }{W}\right) [1-\sqrt{1-F_{L_{1}}(\beta )}].$$

Equation (17) gives the general form of the QZZB for any initial pure state in the photon-loss environment. It is worth noting that such a bound given in Eq. (17) is not fully guaranteed to be tight. Fortunately, with the help of the variational method, the QZZB in the photon-loss environment can be as tight as possible by optimizing the value of the variational parameter [32,48].

Furthermore, by utilizing the inequalities [17]

(18)$$\begin{aligned} 1-\sqrt{1-F_{L_{1}}(\beta )} &\geq \frac{F_{L_{1}}(\beta )}{2},\\ \beta \left( 1-\frac{\beta }{W}\right) &\geq \frac{W}{4}\sin \frac{\pi \beta }{W}, \end{aligned}$$

with the fidelity $F_{L_{1}}(\beta )$ satisfying the conditions of $0\leq F_{L_{1}}(\beta )\leq 1$ and $0\leq \beta \leq W,$ the Eq. (17) can be further rewritten as

(19)$$\sum\nolimits_{Z_{L_{1}}}\geq \sum\nolimits_{Z_{L_{1}}}^{\prime } = \int_{0}^{W}\tilde{F}_{1}(\beta )d\beta ,$$

where $\tilde {F}_{1}(\beta )=\frac {W}{16}F_{L_{1}}(\beta )\sin (\pi \beta /W)$ is denoted as the generalized fidelity under the photon-loss case.

At this asymptotically tight regime, here we shall consider the QZZB for phase estimation problem under the photon losses when inputting three initial states of the probe system $S$, involving the CS (denoted as $\left \vert \psi _{S}\left ( \alpha \right ) \right \rangle$), the SMSVS (denoted as $\left \vert \psi _{S}\left ( r_{1}\right ) \right \rangle$) and the TMSVS (denoted as $\left \vert \psi _{S}\left ( r_{2}\right ) \right \rangle$). Following the approach proposed by Tsang [17], and according to Eq. (19), one can respectively derive the QZZB for phase estimation of the given initial states in the presence of the photon-loss environment, i.e. [see Appendix B for more details],

(20)$$\begin{aligned} \sum \nolimits_{Z_{L_{1}}(\alpha )} &\geq \sum \nolimits_{Z_{L_{1}}(\alpha )}^{\prime } = \frac{\pi ^{3/2}e^{{-}4\eta N_{\alpha }}}{8\sqrt{\eta N_{\alpha }}}\text{erfi}(2\sqrt{\eta N_{\alpha }}),\\ \sum \nolimits_{Z_{L_{1}}(r_{1})} &\geq \sum \nolimits_{Z_{L_{1}}(r_{1})}^{\prime } = \int_{0}^{2\pi }\tilde{F} _{1}(\beta )_{(r_{1})}d\beta ,\\ \sum \nolimits_{Z_{L_{1}}(r_{2})} &\geq \sum \nolimits_{Z_{L_{1}}(r_{2})}^{\prime } = \int_{0}^{2\pi }\tilde{F} _{1}(\beta )_{(r_{2})}d\beta , \end{aligned}$$

where $N_{\alpha }=\left \vert \alpha \right \vert ^{2}$ is the mean photon number of the CS, erfi$(\epsilon )\equiv (2/\sqrt {\pi })\int _{0}^{\epsilon }\exp (t^{2})dt,$ and the generalized fidelities of both the SMSVS and the TMSVS are respectively given by

(21)$$\begin{aligned} \tilde{F}_{1}(\beta )_{(r_{1})} &=\frac{\pi }{8}F_{L_{1}}(\beta )_{(r_{1})}\sin (\beta /2),\\ \tilde{F}_{1}(\beta )_{(r_{2})} &=\frac{\pi }{8}F_{L_{1}}(\beta )_{(r_{2})}\sin (\beta /2), \end{aligned}$$

with

(22)$$\begin{aligned} F_{L_{1}}(\beta )_{(r_{1})} &=\max_{\left \{ \lambda _{1}\right \} }F_{Q_{1}}\left( \beta \right) _{(r_{1})},\\ F_{L_{1}}(\beta )_{(r_{2})} &=\max_{\left \{ \lambda _{1}\right \} }F_{Q_{1}}\left( \beta \right) _{(r_{2})}. \end{aligned}$$

Note that $F_{Q_{1}}\left ( \beta \right ) _{(r_{1})}$=$\left. 1\right / \left \vert \sqrt {1+N_{r_{1}}\left [ 1-\Upsilon ^{2}(\eta,\beta,\lambda _{1})\right ] }\right \vert ^{2}$ with the mean photon number $N_{r_{1}}$=$\sinh ^{2}r_{1}$, and $F_{Q_{1}}\left ( \beta \right ) _{(r_{2})}$=$\left. 1\right / \left \vert 1+\left. N_{r_{2}}\left [ 1-\Upsilon (\eta,\beta,\lambda _{1})\right ] \right / 2\right \vert ^{2}$ with the mean photon number $N_{r_{2}}$=$2\sinh ^{2}r_{2}$ are respectively the lower bound of the fidelity for the SMSVS and the TMSVS, as well as $\Upsilon (\eta,\beta,\lambda _{1})$=$\eta e^{-i\beta }+\left ( 1-\eta \right ) e^{i\beta \lambda _{1}}$. In particular, according to Eq. (20), when $\eta =1$ corresponding to the non-loss case, the corresponding QZZB for the input CS is consistent with the previous work [17].

In order to visually see the effects of photon losses on the QZZB for phase estimation, at a fixed value of $N=5,$ we plot the QZZB $\sum$ as a function of the photon-loss strength $\eta$ for several different states, involving the CS (black dashed line), the SMSVS (red dashed line), and the TMSVS (blue dashed line), as shown in Fig. 1(a). The results show that, with the decrease of $\eta,$ the value of the QZZB for the given states increases. Especially, compared to the another states, the QZZB for the CS increases relatively slowly, which means that the CS as the input is more conducive to reducing the phase estimation uncertainty under the photon losses. Further, to evaluate the gap between the ideal and photon-loss cases, at a fixed $\eta =0.5,$ we also show the QZZB $\sum$ as a function of the mean photon number $N$ for several input resources, i.e., the CS (black lines), the SMSVS (red lines), and the TMSVS (blue lines), as pictured in Fig. 1(b). For comparison, the solid lines correspond to the ideal cases. It is clearly seen that, for the CS, the gap between ideal and photon-loss cases is the smallest, which implies that the CS is more robust against photon losses than other input resources at the same conditions. Moreover, compared with both the CS and the TMSVS, the phase estimation performance of the QZZB for the SMSVS is the worst under the ideal or photon-loss cases. The reasons for these phenomenons can be explained in terms of the generalized fidelity $\tilde {F}_{1}(\beta )$ given in Eq. (19). For this purpose, at fixed values of $N=5$ and $\eta =0.5,$ we consider the generalized fidelity $\tilde {F}_{1}(\beta )$ as a function of $\beta$ for several different initial states, including the CS (black lines), the SMSVS (red lines), and the TMSVS (blue lines), as shown in Fig. 2. According to Eq. (19), the area enclosed by the curve lines and abscissa is the value of the QZZB, which implies that the larger the area, the worse the phase estimation performance. Taking the ideal case as a concrete example, we can clearly see that the area for the SMSVS (red region) is the largest, followed by that for the TMSVS (blue region), and then that for the CS (black region), which is also true for the photon losses.

Fig. 1. The QZZB $\sum$ as a function of (a) the photon-loss parameter $\eta$ with $N=5$ and of (b) the mean photon number $N$ of initial state with $\eta =0.5$. The black, blue and red lines respectively correspond to CS, TMSVS and SMSVS as the initial state. The dashed and solid lines correspond to the photon losses and no photon losses, respectively.

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4. Effects of phase diffusion on the QZZB for phase estimation

More recently, the effects of phase diffusion on the QCRB for phase estimation have been studied in Ref. [48], and they found that compared with photon losses, the existence of phase diffusion has a greater influence on the QCRB. Naturally, the question arises: what are the effects of phase diffusion on the performance of the QZZB for phase estimation? To answer such a question, in this section, we first derive the general form of the QZZB in the presence of phase diffusion by using the variational method and the IWOP technique, and then show the performance of the QZZB with the given states including the $\left \vert \psi _{S}\left ( \alpha \right ) \right \rangle$, the $\left \vert \psi _{S}\left ( r_{1}\right ) \right \rangle$ and the $\left \vert \psi _{S}\left ( r_{2}\right ) \right \rangle$.

Generally speaking, the phase diffusion process can be modeled using the interaction between the probe system $S$ and the environment $E^{\prime }$, which can be described as

(23)$$\exp (i2\kappa \hat{n}\hat{q}_{E^{\prime }})=\exp [i\sqrt{2}\kappa \hat{n}( \hat{a}_{E^{\prime }}+\hat{a}_{E^{\prime }}^{{\dagger} })],$$

where $\kappa$ denotes the strength of phase diffusion and $\hat {q} _{E^{\prime }}=(\hat {a}_{E^{\prime }}+\hat {a}_{E^{\prime }}^{\dagger })/ \sqrt {2}$ is the dimensionless position operator of the mirror. On this background, the density operator $\hat {\rho }_{S+E^{\prime }}^{\prime }\left ( \phi \right )$ in the whole system $S+E^{\prime }$ can be given by

(24)$$\begin{aligned} \hat{\rho}_{S+E^{\prime }}^{\prime }\left( \phi \right) &=\left\vert \Phi _{S+E^{\prime }}(\phi )\right\rangle \left\langle \Phi _{S+E^{\prime }}(\phi )\right\vert\\ &=\hat{U}_{S+E^{\prime }}^{\prime }\left( \phi \right) \hat{\rho} _{S}\otimes \hat{\rho}_{E_{0}^{\prime }}\hat{U}_{S+E^{\prime }}^{\prime \dagger }\left( \phi \right) \end{aligned}$$

where $\hat {\rho }_{S}$ is the same as the aforementioned definition given in Eq. (8), $\hat {U}_{S+E^{\prime }}^{\prime }\left ( \phi \right ) =\exp \left ( -i\phi \hat {n}\right ) \exp (i2\kappa \hat {n}\hat {q}_{E^{\prime }})$ is the corresponding unitary operator of the combined system $S+E^{\prime },$ and $\hat {\rho }_{E_{0}^{\prime }}=\left \vert 0_{E^{\prime }}\right \rangle \left \langle 0_{E^{\prime }}\right \vert$ is the initial state of the phase-diffusion environment, which is often assumed to be the ground state of a quantum oscillator. It is clearly seen from Eq. (24) that the density operator $\hat {\rho }_{S+E^{\prime }}^{\prime }\left ( \phi \right )$ can be viewed as the purifications of the probe state $\hat {\rho }_{S}\left ( \phi \right )$, which is given by [48]

(25)$$\begin{aligned} \hat{\rho}_{S}\left( \phi \right) &=\text{Tr}_{E^{\prime }}\left[ \hat{\rho} _{S+E^{\prime }}^{\prime }\left( \phi \right) \right]\\ &=\sum_{m,n=0}^{\infty }\rho _{m,n}e^{{-}i\phi (m-n)-\kappa ^{2}(m-n)^{2}}\left\vert m\right\rangle \left\langle n\right\vert , \end{aligned}$$

where $\rho _{m,n}=\left \langle m\right \vert \hat {\rho }_{S}\left \vert n\right \rangle$ is the matrix element of the initial state for the probe system $S$. Thus, according to Eq. (24), under the asymptotic condition of $\sqrt {2}\kappa n\gg 1$, the purified unitary evolution can integrally be written as [48]

(26)$$\hat{\rho}_{S+E^{\prime }}^{\prime \prime }\left( \phi \right) =\hat{u} _{E^{\prime }}(\phi )\hat{\rho}_{S+E^{\prime }}^{\prime }\left( \phi \right) \hat{u}_{E^{\prime }}^{{\dagger} }(\phi ),$$

where $\hat {u}_{E^{\prime }}(\phi )=e^{\left. i\phi \lambda _{2}\hat {p} _{E^{\prime }}\right / 2\kappa }$ is the unitary operator with $\lambda _{2}$ being a variational parameter and $\hat {p}_{E^{\prime }}=(\hat {a}_{E^{\prime }}-\hat {a}_{E^{\prime }}^{\dagger })/i\sqrt {2}$ being the dimensionless momentum operator of the mirror, which acts only on the phase-diffusion environment $E^{\prime },$ and connects two purifications $\hat {\rho } _{S+E^{\prime }}^{\prime }\left ( \phi \right )$ and $\hat {\rho }_{S+E^{\prime }}^{\prime \prime }\left ( \phi \right )$ of the same probe state $\hat {\rho } _{S}\left ( \phi \right )$.

Based on the Uhlmann’s theorem [49,50], the fidelity in the phase-diffusion environment $E^{\prime }$ can be given by

(27)$$F_{L_{2}}(\beta )=\max_{\left\{ \left\vert \Phi _{S+E^{\prime }}(\phi )\right\rangle ,\left\vert \Phi _{S+E^{\prime }}(\phi +\beta )\right\rangle \right\} }F_{Q_{2}}(\hat{\rho}_{S+E^{\prime }}^{\prime \prime }\left( \phi \right) ,\hat{\rho}_{S+E^{\prime }}^{\prime \prime }\left( \phi +\beta \right) ),$$

where the maximization is over all purifications $\left \vert \Phi _{S+E^{\prime }}(\phi )\right \rangle$ of $\hat {\rho }_{S+E^{\prime }}^{\prime \prime }\left ( \phi \right )$ and $\left \vert \Phi _{S+E^{\prime }}(\phi +\beta )\right \rangle$ of $\hat {\rho }_{S+E^{\prime }}^{\prime \prime }\left ( \phi +\beta \right )$ in the whole system $S+E^{\prime }$, and $F_{Q_{2}}(\hat {\rho }_{S+E^{\prime }}^{\prime \prime }\left ( \phi \right ) ,\hat {\rho }_{S+E^{\prime }}^{\prime \prime }\left ( \phi +\beta \right ) )\equiv F_{Q_{2}}\left ( \beta \right )$ is the lower bound of the fidelity in the phase-diffusion environment $E^{\prime },$ which can be derived as

(28)$$\begin{aligned} F_{Q_{2}}\left( \beta \right) &=\left\vert \left\langle \Psi _{S+E^{\prime }}(\phi )|\Psi _{S+E^{\prime }}(\phi +\beta )\right\rangle \right\vert ^{2}\\ &=\Theta (\kappa ,\beta ,\lambda _{2})\left\vert \left\langle \psi _{S}\right\vert e^{i\beta \left( \lambda _{2}-1\right) \hat{n}}\left\vert \psi _{S}\right\rangle \right\vert ^{2}, \end{aligned}$$

with $\Theta (\kappa,\beta,\lambda _{2})=e^{\left. -\beta ^{2}\lambda _{2}^{2}\right / 8\kappa ^{2}}$. One can refer to the Appendix C for more details of the corresponding derivation in Eq. (28).

Likewise, if the variational parameter $\lambda _{2}$ takes the optimal value $\lambda _{2opt}$ to make the lower bound of the fidelity as tight as possible, then $F_{Q_{2}}\left ( \beta \right )$ can achieve the maximum value, which is the fidelity in the phase-diffusion environment $F_{L_{2}}(\beta ).$ Further, the lower limit of the minimum error probability $\Pr \nolimits _{e_{L_{2}}}^{el}(\phi,\phi +\beta )$ in the presence of the phase-diffusion environment $E^{\prime }$ can be given by

(29)$$\begin{aligned} \Pr\nolimits_{e_{L_{2}}}^{el}(\phi ,\phi +\beta ) &\geq \frac{1}{2}(1-\frac{ 1}{2}\left\Vert \hat{\rho}_{S+E^{\prime }}^{\prime \prime }\left( \phi \right) +\hat{\rho}_{S+E^{\prime }}^{\prime \prime }\left( \phi +\beta \right) \right\Vert _{1})\\ &\geq \frac{1}{2}[1-\sqrt{1-F_{L_{2}}(\beta )}]. \end{aligned}$$

Similar to how we get the Eq. (19), the QZZB for phase estimation in the presence of the phase-diffusion environment can be denoted as

(30)$$\sum\nolimits_{L_{2}}\geq \sum\nolimits_{Z_{L_{2}}}=\int_{0}^{W}d\beta \frac{ \beta }{2}\left( 1-\frac{\beta }{W}\right) [1-\sqrt{1-F_{L_{2}}(\beta )}],$$

which gives the general form of the QZZB for any initial pure state in the phase-diffusion environment. It is noteworthy that Eq. (30) is also not fully guaranteed to be tight and is only asymptotically tight since the QZZB in the phase-diffusion environment can be as tight as possible by optimizing the value of the variational parameter according to the variational method [32,48].

By using the Eq. (18), finally, the Eq. (30) can be rewritten as

(31)$$\sum \nolimits_{Z_{L_{2}}}\geq \sum \nolimits_{Z_{L_{2}}}^{\prime }=\int_{0}^{W}\tilde{F}_{2}(\beta )d\beta ,$$

where $\tilde {F}_{2}(\beta )=\frac {W}{16}F_{L_{2}}(\beta )\sin (\pi \beta /W)$ is also denoted as the generalized fidelity under the phase-diffusion case.

Likeswise, in this asymptotically tight situation, let us consider the performance of the QZZB for phase estimation under the phase-diffusion environment with the given states including the $\left \vert \psi _{S}\left ( \alpha \right ) \right \rangle$, the $\left \vert \psi _{S}\left ( r_{1}\right ) \right \rangle$ and the $\left \vert \psi _{S}\left ( r_{2}\right ) \right \rangle$. According to the Eq. (31), one can respectively derive the QZZB of these given states in the presence of the phase-diffusion environment, i.e.,

(32)$$\begin{aligned} \sum\nolimits_{Z_{L_{2}}\left( \alpha \right) } &\geq \sum\nolimits_{Z_{L_{2}}\left( \alpha \right) }^{\prime } = \int_{0}^{2\pi }\tilde{F}_{2}(\beta )_{\left( \alpha \right) }d\beta ,\\ \sum\nolimits_{Z_{L_{2}}\left( r_{1}\right) } &\geq \sum\nolimits_{Z_{L_{2}}\left( r_{1}\right) }^{\prime } = \int_{0}^{2\pi }\tilde{F}_{2}(\beta )_{\left( r_{1}\right) }d\beta ,\\ \sum\nolimits_{Z_{L_{2}}\left( r_{2}\right) } &\geq \sum\nolimits_{Z_{L_{2}}\left( r_{2}\right) }^{\prime } = \int_{0}^{2\pi }\tilde{F}_{2}(\beta )_{\left( r_{2}\right) }d\beta , \end{aligned}$$

where we have set

(33)$$\begin{aligned} \tilde{F}_{2}(\beta )_{\left( \alpha \right) } &= \frac{\pi }{8} F_{L_{2}}(\beta )_{\left( \alpha \right) }\sin (\beta /2),\\ \tilde{F}_{2}(\beta )_{\left( r_{1}\right) } &= \frac{\pi }{8} F_{L_{2}}(\beta )_{\left( r_{1}\right) }\sin (\beta /2),\\ \tilde{F}_{2}(\beta )_{\left( r_{2}\right) } &= \frac{\pi }{8} F_{L_{2}}(\beta )_{\left( r_{2}\right) }\sin (\beta /2), \end{aligned}$$

with

(34)$$\begin{aligned} F_{L_{2}}(\beta )_{\left( \alpha \right) } &= \max_{\lambda _{2}}\left\{ \Theta (\kappa ,\beta ,\lambda _{2})\exp \left[{-}2N_{\alpha }\Lambda (\beta ,\lambda _{2})\right] \right\} ,\\ F_{L_{2}}(\beta )_{\left( r_{1}\right) } &= \max_{\lambda _{2}}\frac{ \Theta (\kappa ,\beta ,\lambda _{2})}{\sqrt{1+2N_{r_{1}}\left( 1+N_{r_{1}}\right) \Lambda (2\beta ,\lambda _{2})}},\\ F_{L_{2}}(\beta )_{\left( r_{2}\right) } &= \max_{\lambda _{2}}\frac{ \Theta (\kappa ,\beta ,\lambda _{2})}{1+N_{r_{2}}\left( 1+N_{r_{2}}/2\right) \Lambda (\beta ,\lambda _{2})},\\ \Lambda (\beta ,\lambda _{2}) &=1-\cos \left[ \beta \left( \lambda _{2}-1\right) \right] . \end{aligned}$$

For the sake of clearly seeing the effects of phase diffusion on the QZZB for phase estimation, at a fixed value of $N=5,$ we plot the QZZB $\sum$ as a function of the phase-diffusion strength $\kappa$ for the given states including the CS (black dot-dashed line), the SMSVS (red dot-dashed line) and the TMSVS (blue dot-dashed line), as shown in Fig. 3(a). It is clear that the value of the QZZB for the given states increases with the increase of $\kappa$. In particular, compared to other states, the corresponding QZZB for the SMSVS is relatively larger and increases rapidly as the phase-diffusion strength $\kappa$ increases, which means that both the CS and the TMSVS instead of the SMSVS is a better choice to the robustness against the phase diffusion. More precisely, at the rang of $0\leqslant \kappa \leqslant 0.41,$ the value of the QZZB for the CS can be lower than that for the TMSVS, but the former can be larger than the latter when $\kappa$ is greater than $0.41$. This phenomenon means that the CS can be more sensitive to the phase-diffusion environment compared to the TMSVS [see Fig. 3(b)]. In addition, under the phase diffusion processes (e.g., $\kappa =0.2$), we further consider the QZZB $\sum$ as a function of the mean photon number $N$ for the CS (black lines), the SMSVS (red lines), and the TMSVS (blue lines), as shown in Fig. 3(b). As a comparison, the ideal cases (solid lines) are also plotted here. It is shown that, the gap with the SMSVS (red lines) between the ideal and phase diffusion cases is the largest, which means that the SMSVS is more sensitive to the phase diffusion than other states. We also find that the gap with the TMSVS is smaller than the one with the CS when given the same mean photon number $N$, which does exist in the case of photon losses. Even so, the estimation performance of the QZZB for the CS is superior to that for the TMSVS under the phase diffusion. Similar to Fig. 2, in order to better explain these phenomena, at fixed values of $N=5$ and $\kappa =0.2,$ Fig. 4 shows the generalized fidelity $\tilde {F}_{2}(\beta )$ as a function of $\beta$, in which the area enclosed by the curve lines and abscissa is the value of the QZZB. Likewise, by taking the phase diffusion as an example, the area for the SMSVS (dot-dashed red line) is the largest, followed by that for the TMSVS (dot-dashed blue line), and then that for the CS (dot-dashed black line), which implies that the CS shows the best estimation performance in the presence of the phase diffusion.

Fig. 2. The Generalized fidelity $\tilde {F}_{1}$ as a function of the phase difference $\beta$ with $N=5$ and $\eta =0.5$. The black, blue and red lines respectively correspond to CS, TMSVS and SMSVS as the initial state. The dashed and solid lines correspond to the photon losses and no photon losses, respectively.

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Fig. 3. The QZZB $\sum$ as a function of (a) the phase diffusion parameter $\kappa$ with $N=5$ and of (b) the mean photon number $N$ of initial state with $\kappa =0.2.$ The black, blue and red lines respectively correspond to CS, TMSVS and SMSVS as the initial state. The dot-dashed and solid lines correspond to the phase diffusion and no phase diffusion, respectively.

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Fig. 4. The Generalized fidelity $\tilde {F}_{2}$ as a function of the phase difference $\beta$ with $N=5$ and $\kappa =0.2.$ The black, blue and red lines respectively correspond to CS, TMSVS and SMSVS as the initial state. The dot-dashed and solid lines correspond to the phase diffusion and no phase diffusion, respectively.

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5. Conclusions

In summary, we have used the variational method and the IWOP technique to determine the fidelity in noisy environments by maximizing lower bounds to this quantity, and given a general prescription to perform this maximization. Further, we have also derived the general form of the QZZB for the phase estimation in noisy environments, including the photon loss and the phase diffusion. Both the fidelity and the QZZB in the noisy environments are as tight as possible via the variational method optimizing the value of the variational parameter. In this asymptotically tight situation, we have also presented the performance of the QZZB for phase estimation in noisy systems with several different Gaussian resources, including the CS, the SMSV and the TMSV. It is shown that the performance of the QZZB for phase estimation is related to the generalized fidelity, which can be used to explain the phenomenon that the QZZB for both the CS and the TMSVS performs better than that for the SMSVS, especially for the best performance with inputting the CS. Furthermore, we also investigate the effects of the phase diffusion systems on the QZZB for phase estimation with the same Gaussian resources. It is found that the CS as the initial state can show the better estimation performance of the QZZB at the small range of the phase-diffusion strength ($0\leqslant \kappa \leqslant 0.41$), but for the large range of the phase-diffusion strength $\left ( \kappa >0.41\right ) ,$ its estimation performance can be exceeded by that for the TMSVS.

As a matter of fact, the QZZB has plentiful applications, such as quantum illumination [51,52] and waveform estimation [22]. In particular, Zhuang et al. derived the the ultimate quantum limit on ranging accuracy based on the continuous-time quantum analysis, including the QZZB and QCRB, and they pointed out that quantum illumination ranging provides the mean-squared range-delay precision which is tens of dB better than the case of classical microwave radars [52]. On the other hand, compared with the variational method, how to find a better way to make the QZZB more and more tight in noise environments is still a crucial problem. Based on the above analysises, extending our scheme into these aspects will be our future work.

Appendix A: The Proof of Eqs. (13) and (14)

By using the completeness relation of Fock states, one can obtain

(35)$$\begin{aligned} & \eta ^{\hat{n}}e^{{-}i\beta \hat{n}}=\eta ^{\hat{n}}e^{{-}i\beta \hat{n} }\sum_{k=0}^{\infty }\left\vert k\right\rangle \left\langle k\right\vert =\sum_{k=0}^{\infty }\eta ^{k}e^{{-}i\beta k}\left\vert k\right\rangle \left\langle k\right\vert\\ & =\sum_{k=0}^{\infty }\eta ^{k}e^{{-}i\beta k}\frac{(\hat{a}^{{\dagger} })^{k}}{ k!}\left\vert 0\right\rangle \left\langle 0\right\vert \hat{a}^{k} =\; \colon \; \sum_{k=0}^{\infty }\eta ^{k}e^{{-}i\beta k}\frac{(\hat{a}^{{\dagger} })^{k}}{k!} e^{-\hat{a}^{{\dagger} }\hat{a}}\hat{a}^{k}\colon\\ & =\; \colon \; e^{-\hat{a}^{{\dagger} }\hat{a}}\sum_{k=0}^{\infty }\frac{(\eta e^{{-}i\beta }\hat{a}^{{\dagger} }\hat{a})^{k}}{k!}\colon \; =\; \colon \; \exp \left[ \left( \eta e^{{-}i\beta }-1\right) \hat{n}\right] \colon \end{aligned}$$

where we have utilized the normal ordering form of the vacuum projection operator

(36)$$\left\vert 0\right\rangle \left\langle 0\right\vert =\; \colon \; e^{-\hat{a} ^{{\dagger} }\hat{a}}\colon .$$

Then, using Eq. (35), one can further derive the operator $\hat {Z}$, i.e.,

(37)$$\begin{aligned} \hat{Z}& =\sum_{l=0}^{\infty }\hat{\Pi}_{l}^{{\dagger} }\left( \phi \right) \hat{\Pi}_{l}\left( \phi +\beta \right) =\sum_{l=0}^{\infty }\frac{\left( 1-\eta \right) ^{l}}{l!}\left( \hat{a}^{{\dagger} }\right) ^{^{l}}\eta ^{\hat{n }}e^{{-}i\beta \left( \hat{n}-\lambda _{1}l\right) }\hat{a}^{^{l}}\\ & =\sum_{l=0}^{\infty }\frac{[e^{i\beta \lambda _{1}}\left( 1-\eta \right) ]^{l}}{l!}\left( \hat{a}^{{\dagger} }\right) ^{^{l}}\eta ^{\hat{n}}e^{{-}i\beta \hat{n}}\hat{a}^{^{l}}\\ & =\; \colon \; \sum_{l=0}^{\infty }\frac{[e^{i\beta \lambda _{1}}\left( 1-\eta \right) ]^{l}}{l!}\left( \hat{a}^{{\dagger} }\right) ^{^{l}}\exp \left[ \left( \eta e^{{-}i\beta }-1\right) \hat{n}\right] \hat{a}^{^{l}}\colon\\ & =\; \colon \; \exp \left[ \left( \eta e^{{-}i\beta }-1\right) \hat{n}\right] \sum_{l=0}^{\infty }\frac{[e^{i\beta \lambda _{1}}\left( 1-\eta \right) \hat{ a}^{{\dagger} }\hat{a}]^{l}}{l!}\colon\\ & =\; \colon \; \exp \left[ \left( \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}-1\right) \hat{n}\right] \colon =\left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] ^{\hat{n}}, \end{aligned}$$

where we have utilized the following operator identity about $e^{\lambda \hat {a}^{\dagger }\hat {a}}$ , i.e.,

(38)$$e^{\lambda \hat{a}^{{\dagger} }\hat{a}} =\; \colon \; e^{\left( e^{\lambda }-1\right) \hat{a}^{{\dagger} }\hat{a}}\colon ,$$

to remove the symbol of normal ordering.

Appendix B: The QZZB for the CS in the presence of the photon losses environment

Based on Eq. (15), one can get the lower bound of the fidelity for the CS $\left \vert \psi _{S}\left ( \alpha \right ) \right \rangle$ under the photon-loss environment

(39)$$F_{Q_{1}}\left( \beta \right) _{CS}=\exp \left[ 2N_{\alpha }\left( \eta \cos \beta +\left( 1-\eta \right) \cos \beta \lambda _{1}-1\right) \right] .$$

The value of the variational parameter $\lambda _{1}$ that maximizes the lower bound is $\lambda _{1opt}=0,$ which yields the fidelity for the CS $\left \vert \psi _{S}\left ( \alpha \right ) \right \rangle$, i.e.,

(40)$$F_{Q_{1}}\left( \beta \right) _{CS}=\exp \left[ 2\eta N_{\alpha }\left( \cos \beta -1\right) \right] .$$

Thus, substituting Eq. (40) into Eq. (19), and $W=2\pi$, one can get

(41)$$\sum\nolimits_{Z_{L_{1}}}\geq \sum\nolimits_{Z_{L_{1}}}^{\prime }=\frac{\pi }{8}\int_{0}^{2\pi }\exp \left[ 2\eta N_{\alpha }\left( \cos \beta -1\right) \right] \sin (\beta /2)d\beta ,$$

Changing the integral variable $\beta$ to $s\equiv \cos (\beta /2)$ and utilizing the identity $\cos \beta =2\cos ^{2}(\beta /2)-1,$ one can finally obtain the QZZB for the CS in the presence of the photon-loss environment

(42)$$\sum\nolimits_{Z_{L_{1}}}\geq \sum\nolimits_{Z_{L_{1}}}^{\prime }=\frac{\pi ^{3/2}e^{{-}4\eta N_{\alpha }}}{8\sqrt{\eta N_{\alpha }}}\text{erfi}(2\sqrt{ \eta N_{\alpha }}).$$

Next, using the completeness relation of coherent states, the SMSVS $\left \vert \psi _{S}\left ( r_{1}\right ) \right \rangle$ can be expanded in the basis of the CS,

(43)$$\left\vert \psi _{S}\left( r_{1}\right) \right\rangle =\sqrt{\sec hr_{1}} \int \frac{d^{2}z_{1}}{\pi }e^{-\frac{1}{2}\left\vert z_{1}\right\vert ^{2}+ \frac{1}{2}z_{1}^{{\ast} 2}\tanh r_{1}}\left\vert z_{1}\right\rangle .$$

Then, using Eq. (43), one can obtain the lower bound of the fidelity for the SMSVS in the presence of the photon-loss environment

(44)$$\begin{aligned} & F_{Q_{1}}\left( \beta \right) _{SMSV}=\left\vert \left\langle \psi _{S}\left( r_{1}\right) \right\vert \left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] ^{\hat{n}}\left\vert \psi _{S}\left( r_{1}\right) \right\rangle \right\vert ^{2}\\ & =|\text{sech}r_{1}\int \frac{d^{2}z_{1}d^{2}z_{2}}{\pi ^{2}}\exp [-\frac{1 }{2}(\left\vert z_{1}\right\vert ^{2}+\left\vert z_{2}\right\vert ^{2})\\ & +\frac{1}{2}(z_{1}^{{\ast} 2}+z_{2}^{2})\tanh r_{1}]\times \left\langle z_{2}\right\vert \colon \; \exp \left[ \left( \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}-1\right) a^{{\dagger} }a\right] \colon \; \left\vert z_{1}\right\rangle |^{2}\\ & =|\text{sech}r_{1}\int \frac{d^{2}z_{1}d^{2}z_{2}}{\pi ^{2}}\exp [-\frac{1 }{2}(\left\vert z_{1}\right\vert ^{2}+\left\vert z_{2}\right\vert ^{2})\\ & +\left( \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}-1\right) z_{1}z_{2}^{{\ast} }\\ & +\frac{1}{2}(z_{1}^{{\ast} 2}+z_{2}^{2})\tanh r_{1}]|^{2}\\ & =\frac{1}{\left\vert \sqrt{1+N_{1}\left[ 1-\Upsilon ^{2}(\eta ,\beta ,\lambda _{1})\right] }\right\vert ^{2}}, \end{aligned}$$

where we have used the Eq. (38) and the integral formular

(45)$$\int \frac{d^{2}\gamma }{\pi }\exp (\varsigma \left\vert \gamma \right\vert ^{2}+\xi \gamma +\omega \gamma ^{{\ast} }+f\gamma ^{2}+g\gamma ^{{\ast} 2})= \frac{1}{\sqrt{\varsigma ^{2}-4fg}}\exp \left( \frac{-\varsigma \xi \omega +\xi ^{2}g+\omega ^{2}f}{\varsigma ^{2}-4fg}\right) .$$

Likewise, by using the completeness relation of Fock states, one can get the TMSVS $\left \vert \psi _{S}\left ( r_{2}\right ) \right \rangle$ can be expanded in the basis of Fock states, i.e.,

(46)$$\left\vert \psi _{S}\left( r_{2}\right) \right\rangle =\sec hr_{2}\sum_{n=0}^{\infty }\left( -\tanh r_{2}\right) ^{n}\left\vert n,n\right\rangle .$$

Then, utilizing the Eq. (46), one can finally obtain the lower bound of the fidelity for the TMSVS in the presence of the photon-loss environment

(47)$$\begin{aligned} & F_{Q_{1}}\left( \beta \right) _{TMSV}\\ & =\left\vert \left\langle \psi _{S}\left( r_{2}\right) \right\vert \left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] ^{ \hat{n}}\sec hr_{2}\sum_{n=0}^{\infty }\left( -\tanh r_{2}\right) ^{n}\left\vert n,n\right\rangle \right\vert ^{2}\\ & =\left\vert \sec h^{2}r_{2}\sum_{n=0}^{\infty }\left[ \tanh ^{2}r_{2}\left[ \eta e^{{-}i\beta }+\left( 1-\eta \right) e^{i\beta \lambda _{1}}\right] \right] ^{n}\right\vert ^{2}\\ & =\frac{1}{\left\vert 1+\left. N_{r_{2}}\left[ 1-\Upsilon (\eta ,\beta ,\lambda _{1})\right] \right/ 2\right\vert ^{2}}. \end{aligned}$$

Appendix C: The Proof of Eq. (28)

Using the completeness relation of momentum states,

(48)$$\int_{-\infty }^{\infty }dp_{E^{\prime }}\left\vert p_{E^{\prime }}\right\rangle \left\langle p_{E^{\prime }}\right\vert =1,$$

where $\left \vert p_{E^{\prime }}\right \rangle$ is the eigenstate of the momentum operator $\hat {p}_{E^{\prime }}$ [36,38]

(49)$$\left\vert p_{E^{\prime }}\right\rangle =\pi ^{{-}1/4}\exp \left( -\frac{1}{2} p_{E^{\prime }}^{2}+i\sqrt{2}p_{E^{\prime }}\hat{a}_{E^{\prime }}^{{\dagger} }+ \frac{\hat{a}_{E^{\prime }}^{{\dagger} 2}}{2}\right) \left\vert 0_{E^{\prime }}\right\rangle ,$$

one can obtain

(50)$$\begin{aligned} & \left\langle \Psi _{S+E^{\prime }}(\phi )|\Psi _{S+E^{\prime }}(\phi +\beta )\right\rangle\\ & =\left\langle \Phi _{S+E^{\prime }}(\phi )\right\vert e^{\left. -ix\lambda _{2}\hat{p}_{E^{\prime }}\right/ 2\kappa }e^{\left. i(x+\beta )\lambda _{2} \hat{p}_{E^{\prime }}\right/ 2\kappa }\left\vert \Phi _{S+E^{\prime }}(\phi +\beta )\right\rangle\\ & =\left\langle \psi _{S}\right\vert e^{i\beta (\lambda _{2}-1)\hat{n} }\left\vert \psi _{S}\right\rangle \left\langle 0_{E^{\prime }}\right\vert e^{\left. i\beta \lambda _{2}\hat{p}_{E^{\prime }}\right/ 2\kappa }\left\vert 0_{E^{\prime }}\right\rangle\\ & =\left\langle \psi _{S}\right\vert e^{i\beta (\lambda _{2}-1)\hat{n} }\left\vert \psi _{S}\right\rangle \left\langle 0_{E^{\prime }}\right\vert e^{\left. i\beta \lambda _{2}\hat{p}_{E^{\prime }}\right/ 2\kappa }\int_{-\infty }^{\infty }dp_{E^{\prime }}\left\vert p_{E^{\prime }}\right\rangle \left\langle p_{E^{\prime }}|0_{E^{\prime }}\right\rangle\\ & =e^{\left. -\beta ^{2}\lambda _{2}^{2}\right/ 16\kappa ^{2}}\left\langle \psi _{S}\right\vert e^{i\beta (\lambda _{2}-1)\hat{n}}\left\vert \psi _{S}\right\rangle , \end{aligned}$$

where we have utilized the integrational formula

(51)$$\int_{-\infty }^{\infty }\exp ({-}hy^{2}+gy)dy=\sqrt{\pi /h}\exp (g^{2}/4h).$$

Therefore, the lower bound of the fidelity in the phase diffusion environment can be given by

(52)$$F_{Q_{2}}\left( \beta \right) =\Theta (\kappa ,\beta ,\lambda _{2})\left\vert \left\langle \psi _{S}\right\vert e^{i\beta \left( \lambda _{2}-1\right) \hat{n}}\left\vert \psi _{S}\right\rangle \right\vert ^{2}.$$

Funding

National Natural Science Foundation of China (11534008, 11664017, 11964013, 62161029, 91536115); Natural Science Foundation of Shaanxi Province (2016JM1005); the Training Program for Academic and Technical Leaders of Major Disciplines in Jiangxi Province (20204BCJL22053); Natural Science Foundation of Jiangxi Provincial (20202BABL202002).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

The basic data of the results of this paper have not been published, but can be obtained from the author according to reasonable requirements.

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Evaluating the quantum Ziv–Zakai bound for phase estimation in noisy environments

Abstract

1. Introduction

2. QZZB

3. Effects of photon losses on the QZZB for phase estimation

4. Effects of phase diffusion on the QZZB for phase estimation

5. Conclusions

Appendix A: The Proof of Eqs. (13) and (14)

Appendix B: The QZZB for the CS in the presence of the photon losses environment

Appendix C: The Proof of Eq. (28)

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (52)

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