1. Introduction

Phase estimation [1] is an important issue in quantum metrology, which has been widely studied in various types of optical interferometers, such as the Mach-Zehnder interferometer, SU(1,1) interferometer [2], and other modified interferometers [3,4]. Nonclassicality and entanglement are considered to be the main resources for improving the phase sensitivity of interferometers. Previous research has demonstrated that the phase sensitivity of interferometers can be improved by inputting nonclassical [5]or entangled quantum states [6]. Nonclassical quantum states are considered to have great potential in the field of quantum metrology, and there are numerous approaches for quantifying nonclassicality, such as variance [7], metrological power [8] and the negativity of the quasiprobability distribution [9]. The nonclassicality state may be produced by a classical state by appropriately modifying the measurement [10], or by applying non-Gaussian operators such as photon-addition or photon-subtraction on the initial states. Quantum entanglement also plays a central role in phase estimation and quantum information processing [11] and an entangled quantum state has been experimentally generated [12]. The entanglement of formation [13] and concurrence [14] are two well-defined quantitative measures of quantum entanglement, and the correlation between the quantum Fisher information and quantum entanglement is discussed for the correlated two-mode field state in Ref. [15]. In addition, it is well known that nonlinear effects are an important resource for enhancing the accuracy of phase estimation [16]. These results indicate that the phase sensitivity can be greatly improved by using nonlinear effects, such as nonlinear phase shifters [17] and nonlinear optical elements [18,19]. It has been shown in numerous studies that a nonlinear modified interferometer has better sensitivity than traditional LI [20,21]. However, super-resolution detection, such as parity detection, is also considered as one of the main methods for improving the phase sensitivity of interferometers [22]. Mathematically, the parity signal can be obtained by calculating the Wigner function of the output state [23] or a new operator method [24], and a parity detection scheme has also been experimentally demonstrated [22]. Correspondingly, in this study, by combining the properties of entanglement and the nonclassicality of quantum states, nonlinear interaction and parity detection, we propose a new interferometer model to improve phase estimation and analyze the phase sensitivity of two interferometers with different input states. Furthermore, we quantitatively study the effect of nonclassicality and entanglement on phase sensitivity of LI and NI. Our results demonstrated that the phase sensitivity of interferometers can be significantly improved by enhancing the nonclassical or entanglement properties of quantum states.

The remainder of this report is organized as follows. In Section 2, we review the notion of quantum Fisher information (QFI), nonclassicality and entanglement, and give the analytic expressions of the QFI of several common quantum states. It is worth mentioning that a new expression of the nonclassicality quantifier for a two-mode quantum state is obtained by extending the quantifier for a single-mode quantum state. The nonclassicality of four different quantum states is then obtained using the new method. We also specify the amount of entanglement of Fock states (EFS) and entangled coherent state (ECS) in terms of concurrence, and analyze the relationship between the QFI and the amount of entanglement. In Section 3, we focus on the phase sensitivity of two interferometers (linear and nonlinear) with different input states: (i) We obtain an analytical expression for the parity signal and phase sensitivity of LI. (ii) For NI, we introduce a new operator method for solving the expectation of the parity operator. Based on the techniques of integration within an ordered product of operators (IWOP) and coherent state representation theory, the analytical expression for the parity signal and the phase sensitivity of LI and NI are also determined. The effect of entanglement and noncalssicality on the phase sensitivity of two interferometers is discussed in section 4. Finally, Section 5 presents the main conclusions.

2. Measure of nonclassicality and entanglement for two-mode quantum states

Let us briefly review the theoretical model of phase estimation [1,25], which includes three processes (preparation of the input state, unitary evolution, and detection). For the input state $\left \vert \psi \right \rangle _{in},$ after it passes through a phase shifter, the output state $\left \vert \Psi \right \rangle _{out}$ becomes $U(\phi )\left \vert \psi \right \rangle _{in}$. For a single parameter $\theta$ estimation, the precision of parameter estimation can be analyzed by quantum Fisher information (QFI) and quantum Cramer–Rao bound, i.e. var$(\theta )\geq \frac {1}{vF}$, where var$(\theta )$ is the variance of $\theta$ and $v$ is the number of independent measurements [26]. For a parameterized quantumstate $\rho _{\theta }$, the quantum Fisherin formation $F$ is defined as $F=tr(\rho _{\theta }L^{2})$, where L is the so-called symmetric logarithmic derivative operator which has been described in detail in Ref. [27]. The QFI describes the information associated with phase estimation limited only by the initial quantum state. For a input pure state, the QFI can be calculated by using the following formula [28]

For a single mode quantum state, the nonclassicality $N(\rho )$ can be quantified using the variance $N(\rho )=\int V_{\left \vert z\right \rangle \left \langle z\right \vert }(\rho )\frac {d^{2}z}{\pi },$ which can be expressed as $N(\rho )=\int \left \langle z\right \vert \rho ^{2}\left \vert z\right \rangle -\left \langle z\right \vert \rho \left \vert z\right \rangle ^{2} \frac {d^{2}z}{\pi }$. Moreover, among pure states, $N(\rho )$ achieves a minimum value of $1/2$ if and only if $\rho =\left \vert \alpha \right \rangle \left \langle \alpha \right \vert$ is a coherent state. If $N(\rho )>1/2$ , then the state $\rho$ must be nonclassical. For example, for any Fock state, $N(\left \vert n\right \rangle \left \langle n\right \vert )=1-C_{2n}^{n} \frac {1}{2^{2n+1}}\geq 1/2,$ which shows that all Fock states ($n\geq 1$) are nonclassical. Actually, Fock states are the most nonclassical states compared to other quantum states with the same average photon number.

We may extend the notion of the nonclassicality quantifier to a two-mode quantum state as follow

This nonclassicality quantifier $N(\rho )$ has been given an intuitive interpretation and properties in Ref. [7]. Eq. (2) characterizes the purely quantum phase-space localization of the state. This nonclassicality quantifier $N(\rho )$ for two-mode quantum states also has the following properties:

Similarly $,$ for the two-mode product Fock state ($PFS$)$\ \rho =\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{bb}\left \langle n\right \vert \left \langle m\right \vert _{a}$, the nonclassicality quantifier $N(\rho )$ can be calculated as

Several studies have shown that the quantum entanglement is very useful for improving the accuracy of phase estimation. However, excessive entanglement should be avoided because it adversely affects sensitivity. Most of the previous research on the effect of entanglement to phase sensitivity of interferometers were qualitative studies. In this report, we quantitatively investigate the effect of entanglement on the phase sensitivity of an interferometer by considering concurrence. The entanglement of a two-mode quantum state can be measured in terms of the von Neumann entropy [29] and the concurrence function [14,30]. Currently, it is generally accepted that concurrence can be used to measure entanglement.

For a maximally entangled Fock state, i.e. $EFS$ state ($\left \vert \Psi \right \rangle _{\text {EFS}}=\frac {1}{\sqrt {2}}(\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b}+\left \vert n\right \rangle _{a}\left \vert m\right \rangle _{b}),$ and when $m=0$, $EFS$ reduce to $N00N$) and two-mode entangled coherent states ($ECS$) ($\left \vert \Psi \right \rangle _{\text {ECS }}=N_{\alpha \beta }(\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b}+\left \vert \beta \right \rangle _{a}\left \vert \alpha \right \rangle _{b}),$ $N_{\alpha \beta }^{2}=\frac {1}{2(1+e^{-\left \vert \alpha -\beta \right \vert ^{2}})},$ and when $\beta$=0, $ECS$ reduce to $ecs$, also known as the entangled coherent states), and the concurrence of $EFS$ and $ECS$ are calculated as follows

This result can also be found in In Ref. [31]. In addition, according to Eq. (2), we can also obtain the nonclassicality quantifier $N(\rho )$ as

In particular, to study the relationship among QFI, entanglement and nonclassicality, we consider a particular input state, i.e. ecs, $\left \vert \Psi \right \rangle _{\text {ecs}}=N_{\alpha }(\left \vert 0\right \rangle _{a}\left \vert \alpha \right \rangle _{b}+\left \vert \alpha \right \rangle _{a}\left \vert 0\right \rangle _{b},$ which is presented in in Ref. [32]. The quantum fisher information and total average photon number of ecs are follows

We can solve $\left \vert \alpha \right \vert ^{2}$ in Eqs. (4) and (5) such that $\left \vert \alpha \right \vert ^{2}$ are functions of $N$ and $E$. By substituting $\left \vert \alpha \right \vert ^{2}=\ln (\frac {8N-7}{5-8N+2 \sqrt {2}\sqrt {4N-3}})$, $N_{\alpha }^{2}=\frac {8N-7}{4(\sqrt {2}\sqrt {4N-3}-1) }$, and $\left \vert \alpha \right \vert ^{2}=Ln(\frac {1+E}{1-E})$, $N_{\alpha }^{2}=\frac {1+E}{4}$ in $F_{\text {ECS}}$ of Eq. (6), we find that the quantum Fisher information $F_{\text {ecs}}$ can be expressed as a function of the concurrence $E$ or nonclassicality $N$ as follows

By analyzing Eqs. (3) and (5), we find that: $N(\rho _{\text {PCS}})=0.75,$ $0.75<N(\rho _{\text {ECS}})<0.875,$ $0.875$ $(n=1)\leq N(\rho _{N00N})<1$ and $0.875$ $(n=1)\leq$ $N(\rho _{\text {PFS}})<1.$ Further, we also plot the nonclassicality $N(\rho )$ as a function of the total average photon number the $N_{T}$ in Fig. 1. It is evident that the relationship between the four different nonclassicality input states is $N(\rho _{\text {C}})<N(\rho _{ \text {ECS}})<N(\rho _{\text {PFS}})<N(\rho _{N00N}).$ As such, twin Fock states, ECS and N00N are all nonclassical states, but N00N is more nonclassical than the other three states if the total average photon number is the same. Theoretically, for input ecs, both entanglement and nonclassicality improve the accuracy of phase estimation (see Fig. 2).

 

Fig. 1. The nonclassicality $N(\rho )$ verse total average photon number $N_{T}$ for four different input states, produce coherent states $\left \vert \Psi \right \rangle _{\text {PCS}}$, product Fock states $\left \vert \Psi \right \rangle _{\text {PFS}}$, maximally entangled states $\left \vert \Psi \right \rangle _{N00N}$ and entangled coherent state $\left \vert \Psi \right \rangle _{\text {ECS}}$. Black solid line: $PCS$; Red line: $ECS$; Orange line: $PFS$; Blue line: $N00N$. The total average photon number of the input states $N_{T}$ can be calculated as $N_{T}=\left \langle a^{\dagger }a+b^{\dagger }b\right \rangle _{in}$. For a valid comparison, all input states are implicitly parametrized by the total average photon number.

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Fig. 2. The optimal phase estimation for ecs given by the Quantum Cramer-Rao bound $\delta \phi \geq \frac {1}{\sqrt {F_{ \text {ecs}}}}$ as a function of the concurrence $E$ (left) and nonclassicality $N$ (right). The precision of phase estimation increases with the increase of entanglement and nonclassicality.

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In the following section, we mainly discuss the phase sensitivity of linear and nonlinear interferometer with different input states, i.e., single-mode Fock state and coherent state, two-mode product Fock state and coherent state, and two-mode entangled Fock state and two-mode entangled coherent state. Moreover, we quantified the effect of the nonclassicality and entanglement on phase sensitivity of linear and nonlinear interferometers.

3. Phase sensitivity of linear and nonlinear interferometers with parity detection

Optical interferometers, such as the MZI and SU(1,1)I [33], are among the most common tools for studying the phase sensitivity of interferometers. For the past few decades, numerous studies have confirmed confirmed that the phase sensitivity of an interferometer can be greatly improved by exploiting exploiting the non classical properties [34] and the entanglement [35] of quantum states or nonlinear effects [16]. At present, investigations are mainly focused on the reconstruction of traditional interferometers [20,21,36]. The results have shown that reconstructed interferometers are often robust to photon number loss. In this report, we propose a new scheme involving linear and nonlinear interferometers (see Fig. 3).

 

Fig. 3. Sketch of a modified interferometer with $\left \vert \psi \right \rangle _{i}$ as initial input state. The interferometer’s input state $\left \vert \Psi \right \rangle _{i}$ is generated after the initial input states $\left \vert \psi \right \rangle _{i}$ impinges on a magic beam splitter (MBS). After applying a phase shifter $U(\phi )$ to the $b$ mode of the path, the optical components are combined in a specific optical element (SOE), and become the final output state $\left \vert \Psi \right \rangle _{o}$. Here, the SOE may be a beam splitter (BS) or an optical parametric amplifier (OPA), which represents a linear or nonlinear interferometer, respectively.

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Phase sensitivity is a physical quantity for measuring the accuracy of phase estimation. The primary objective of interferometry is to estimate phase shifts with high sensitivity The phase sensitivity ($\Delta \phi )$ of an interferometer can be characterized using the following error propagation formula:

3.1 Phase sensitivity of LI with different input states

The LI is composed of a magic beam splitter [22], a conventional beam splitter and a phase shifter $U(\phi )=e^{ib^{\dagger }b\phi }$. Here, the magic beam splitter can generate arbitrary input states, and for the second beam splitter, we select a special beam splitter that can be represented as $U_{BS}=e^{-i\frac {\pi }{4}(a^{\dagger }b+ab^{\dagger })}$. The relationship between the input operators ($a_{i}$, $b_{i}$) and output operators ($a_{o}$, $b_{o}$) is given by

In this report, all input states ($\left \vert \Psi \right \rangle _{i}$) of the interferometers are regarded as the states after initial input states ($\left \vert \psi \right \rangle _{i}$) pass the first optical device, which is a magic beam splitter can be generated different input states. For example, input states, i.e. two-mode product coherent state ($PCS$) ($\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b})$ can be generated after $\left \vert \alpha ^{^{\prime }}\right \rangle _{a}\left \vert \beta ^{^{\prime }}\right \rangle _{b}$ ($\alpha ^{^{\prime }}=\frac {\alpha +\beta }{\sqrt {2}},$ $\beta ^{^{\prime }}=\frac {\alpha -\beta }{\sqrt {2}}$) enters a special 50:50 beam splitter $U_{BS}=e^{i\frac {\pi }{4}(a^{\dagger }b+ab^{\dagger })}.$ After the phase shift (taken to be in the $b$-mode), the input states become $\left \vert \Psi _{\phi }\right \rangle =\left \vert \alpha \right \rangle _{a}\left \vert \beta e^{i\phi }\right \rangle _{b}$, which carry information about phase $\phi$. Finally, the output states become $\left \vert \Psi _{out}\right \rangle =\left \vert \frac {(\alpha +\beta e^{i\phi })}{\sqrt {2}}\right \rangle _{a}\left \vert \frac {(-\alpha +\beta e^{i\phi })}{\sqrt {2}}\right \rangle$ after the second specific optical element. As a matter of fact, the phase-matching condition is an essential step in the search for the optimal probe state, as discussed in Ref. [37]. In order to better study the effect of entanglement and noncalssicality on phase sensitivity of interferometers, the optimal phase angle of the initial input states is set at zero.

The parity operator is defined as $\Pi _{b}=e^{i\pi b^{\dagger }b}$, and the expectation value $\left \langle \Pi _{b}\right \rangle$ of the parity operator and the partial derivative for input $PCS$ can be calculated as follows

Similarly, the product Fock state ($PFS$) $\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b}$ can be generated by a magic optical device that transforms the initial state into input states $\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b}$. After a series of calculations, we finally find that $\left \langle \Pi _{b}\right \rangle _{ \text {PFS}}=C$ ($C$ is a constant), which signifies that the phase information is not included in the detected signal. As such, under parity detection, we cannot recover any phase information when the input states are $PFS.$

Let us consider an entangled Fock states ($EFS$), i.e. $\left \vert \Psi _{ \text {EFS}}\right \rangle =\frac {1}{\sqrt {2}}(\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b}+\left \vert n\right \rangle _{a}\left \vert m\right \rangle _{b}).$ It can be produced, for example, by post-selecting the output of a pair of optical parametric oscillators [38]. Finally, the parity signal can be calculated as

The EFS reduces to a $N00N$ state when $m=0$, which can be produced after an initial state $\left \vert n\right \rangle _{a}\left \vert 0\right \rangle _{b}$ gets through an magic optical device. The expectation value of the parity operator is reduced to $\left \langle \Pi _{b}\right \rangle =(-1)^{n}\cos n\phi,$ and consequently $\Delta \phi =1/n.$

When considering $ECS$ ($\left \vert \Psi _{\text {ECS}}\right \rangle =N_{\alpha }(\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b}+\left \vert \beta \right \rangle _{a}\left \vert \alpha \right \rangle _{b}))$ as input states, and $\alpha$ and $\beta$ are both real numbers, we can obtain the expectation value and the partial derivative of $\left \langle \Pi _{b}\right \rangle _{\text {ECS}}$ as follows

When $\beta =0,$ the ECS reduces to ecs, i.e. $\left \vert \Psi _{ \text {ecs}}\right \rangle =N_{\alpha }(\left \vert \alpha \right \rangle _{a}\left \vert 0\right \rangle _{b}+\left \vert 0\right \rangle _{a}\left \vert \alpha \right \rangle _{b}),$ which can be generated by in putting a coherent state ($\left \vert \alpha /\sqrt {2}\right \rangle _{a}$) in mode $a$ and an even coherent state $N_{\alpha }(\left \vert \alpha /\sqrt {2}\right \rangle _{b}+\left \vert -\alpha /\sqrt {2}\right \rangle _{b})$ in mode $b$ into the first 50:50 BS. Then, Eq. (12) reduces to

Substituting Eqs. (10), (11), and (13) into Eq. (8), we obtain the phase sensitivity of MZI with input states $PCS$, $N00N$ and $ecs$ as follows

According to Eq. (14), we plot the phase sensitivity as a function of total average photon number $N_{T}$ in Fig. 4. It is evident that the relationship for the phase sensitivity of LI is: $\Delta \phi _{\text {ecs}}<\Delta \phi _{ \text {N00N}}$ $<\Delta \phi _{\text {PCS}}.$ Thus, the phase sensitivity of a linear interferometer with ecs as the inputs is better than that of $N00N$ or with coherent state as the inputs, and it exceeds the Heisenberg limit (1/N) which can be achieved by $N00N$. These results are consistent with those in Ref. [32].

 

Fig. 4. Phase sensitivity of linear interferometer as a function of total average photon number $N_{T}$ with PCS, ecs and N00N as inputs. Red dotted line: $ecs$; Blue line: $N00N$; Black solid line: PCS.

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3.2 Phase sensitivity of NI with different input states

When the right SOE is an optical parametric amplifier (OPA), an NI is formed (see Fig. 3), which is composed of a MBS and an OPA. It can also be treated as a variation of the SU(1,1) interferometer which is composed of two OPAs. The operator form of the OPA [39]can be described by a two-mode squeezing operator $U_{OPA}=e^{\xi a^{\dagger }b^{\dagger }-\xi ^{\ast }ab}$ with squeezing parameter $\xi =ge^{i\theta }$, where $g$ and $\theta$ are the gain factor and the phase shift of the parametric process, respectively.

3.2.1 Normal ordered form of the nonlinear interferometer’s evolution operator $U(\xi,\phi )$ and detection operator $u(\xi,\phi )$

Based on IWOP, we can determine that the normal ordered form of OPA operator [40] $U_{OPA}$ is

According to Ref. [42], the unitary transformation associated with the NIr can be represented by the following unitary operator

The parity operator $\Pi _{b}$ is defined as $\Pi _{b}=(-1)^{b^{\dagger }b}=e^{i\pi b^{\dagger }b},$ and it can be written as $\Pi _{b}=e^{i\pi b^{\dagger }b}=\int \frac {d^{2}\beta }{\pi }\left \vert \beta \right \rangle _{b}\left \langle -\beta \right \vert$ in the coherent state representation. Therefore, we can easily obtain the value of the expectation of the parity operator as

After a complex, we finally obtain the normal ordered form of $u(\xi,\phi )$ as

Hitherto, we obtained the normal ordered form of the Hermitian operator $u(\xi,\phi )$, and the parity signal can be obtained by determining the expectation of the parity operator in the input states. By recasting the expression for the input quantum states in the coherent state representation and using the eigenvalue equations of the annihilation operator $a\left \vert \alpha \right \rangle =\alpha \left \vert \alpha \right \rangle$, we can easily calculate the expectation value of the parity operator based on Eqs. (20), (22) and (23). In the following, we will derive the parity signal for several given input states, and give the analytical expression of the phase sensitivity of NI.

3.2.2 Phase sensitivity of NI with different input states

Nonlinear interferometers [33,43] can potentially improve phase sensitivity and surpasses standard quantum limit using only classical input states [19,44]. In this section, we express the new form of several common input states ($\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b}$, $\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b},$ $N_{\alpha \beta }(\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b}+\left \vert \beta \right \rangle _{a}\left \vert \alpha \right \rangle _{b})$ ($N_{\alpha \beta }^{2}=\frac {1}{ 2(1+e^{-\left \vert \alpha -\beta \right \vert ^{2}})}$) and $\frac {1}{\sqrt {2} }(\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b}+\left \vert n\right \rangle _{a}\left \vert m\right \rangle _{b})$) in the coherent representation, and calculate the average value of the parity operator. When the input state is $PCS$ ($\left \vert \alpha \right \rangle _{a}\left \vert \beta \right \rangle _{b})$, combining Eqs. (20) and (22), we can obtain the average value of parity operator $\Pi _{b}$ as follows

We consider a typical type of Gaussian input state, i.e. Fock state, which can be expressed in the coherent representation as

So, for input $PFS$ ($\left \vert m\right \rangle _{a}\left \vert n\right \rangle _{b})$ and $EFS$ $(\frac {1}{\sqrt {2}}(m_{a}\left \vert n\right \rangle _{b}+\left \vert n\right \rangle _{a}\left \vert m\right \rangle _{b})),$ we can easily obtain the average value of the parity operator ($\left \langle \Pi \right \rangle _{b}$) after a series of calculations. It can be expressed as

Next, we focus on the inputs ECS and EFS. Similarly, the expectation of the parity operator can also be immediately obtained as

Combining Eqs. (24), (27) and (8), we obtain the numerical expression for the phase sensitivity of the NI, and plot this parameter as a function of the total average photon number $N_{T}$ with PCS and ECS as the input states, as shown in Fig. 5. It is evident that the phase sensitivity with PCS (no entanglement) as inputs is lower than that of ECS for the same total average photon number. It is worth mentioning that the phase sensitivity of the ECS exceeds the Heisenberg limit (1/N), whereas PCS does not achieve the Heisenberg limit Even for NI. These results strongly suggest that entanglement can greatly improve the phase sensitivity of NI.

 

Fig. 5. Phase sensitivity of NI as a function of the total average photon number $N_{T}$ with PCS and ECS as input states. Black dotted line: $\Delta \phi _{\text {PCS}}$; Red dashed line: $\Delta \phi _{\text { ECS}};$ Blue line: Heisenberg limit $1/N$, where, $\beta =\frac { 0.5+0.5i}{\sqrt {2}},$ $\theta =0,$ and $g$ and $\phi$ are set at the optimal point.

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4. Discussion: the phase sensitivity of LI and NI verse nonclassicality and entanglement

There have been numerous qualitative studies on the effects of entanglement degree and the non-classical properties of quantum states on the phase sensitivity of interferometers [5,6,15]. However, there are few quantitative studies on the effects of entanglement and noncalssicality on phase sensitivity. Based on the results obtained in this investigation, two kinds of important problems are explored in this section: $(i)$ the effect of entanglement and nonclassicality on the phase sensitivity of interferometers; $(ii)$ the process by which entanglement and noncalssicality of quantum states affect the phase sensitivity of two interferometers.

Firstly, we plot the phase sensitivity of two interferometers as a function of entanglement and nonclassicality according to Eqs. (14), (24) and (27), respectively. The phase sensitivity can be expressed as a function of the nonclassicality $N$ for $N00N$ and ecs by substituting $\left \vert \alpha \right \vert ^{2}$ using $E$ or $N$ in Eq. (14). We then also also plot the phase sensitivity of LI as a function of the $E$ and $N$ in Fig. 6. It is evident that both entanglement and nonclassicility can improve the phase sensitivity of LI for the same input state. It should be noted that the nonclasscality is not a requirement for improving the phase sensitivity of a LI. For example, the nonclassicality of ECS and N00N is: $N(\rho _{ \text {ecs}})<N(\rho _{N00N})$, but the phase sensitivity with ecs and N00N is:$\ \Delta \phi _{\text {ecs}}<\Delta \phi _{N00N}$ (see Fig.6 (righ) ). It should also be noted that: $(i)$ The phase sensitivity with two-mode product coherent states is independent of $N_{\text {C}}$ ($N_{\text {c}}=\frac {3}{4}$). $(ii)$ The phase sensitivity with $N00N$ is independent of $E_{N00N}$ ($E_{N00N}=1$). By substituting $\left \vert \alpha -\beta \right \vert ^{2}=\ln (\frac {8N-7}{5-8N+2\sqrt {2}\sqrt {4N-3}})$, $N_{\alpha }^{2}=\frac {8N-7}{4( \sqrt {2}\sqrt {4N-3}-1)}$, and $\left \vert \alpha -\beta \right \vert ^{2}=\ln (\frac {1+E}{1-E})$, $N_{\alpha }^{2}=\frac {1+E}{4}$ in $\left \langle \Pi _{b}\right \rangle _{\text {ECS}}$ of Eq. (27) and combining Eq. (8), we can rewrite the phase sensitivity of NI as a function of the concurrence $E$ or nonclassicality $N$. These analytic results are shown in Fig. 7.

 

Fig. 6. Phase sensitivity of MZI as a function of the entanglement $E$ (left) with ecs as inputs, and nonclassicality $N$ (right) with ecs and $N00N$ inputs. Red line: $ecs$; Blue line: $N00N.$

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Fig. 7. Phase sensitivity of NI as a function of the concurrence $E$ (left) and nonclassicality $N$ (right) with ECS.

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To examine the first question, we can compare the phase sensitivity of two interferometers in the case of a single-mode ($\left \vert \alpha \right \rangle$) or a two-mode product quantum state ($\left \vert \alpha \right \rangle \left \vert \beta \right \rangle$) and a two-mode entangled quantum state (ECS) as the input states. Based on Eqs. (8), (14) and (27), the phase sensitivity $\Delta \phi _{\text {PCS}}$ and $\Delta \phi _{\text { ECS}}$ are plotted in Figs. (6) and (7). To investigate the specific effect of entanglement on the phase sensitivity of two interferometers (second question), we can examine the phase sensitivity of two interferometers with EFS and ECS as inputs, as shown in the left of Figs. (6) and (7). The effect of noncalssicality on the phase sensitivity of two interferometers can be investigated with any noncalssical quantum states, and numerical results are presented in the right of Figs. (6) and (7).

By analyzing these numerical and simulation results, we we arrive at the following three important results: $(i)$ compared to a single mode quantum state and a double mode direct product quantum state, the entangled quantum state can improve the phase sensitivity of the interferometer for the same input energy condition (see Fig. 7). In addition, the phase sensitivity of two interferometers can be improved by increasing the entanglement for the same entangled quantum state (see Fig. 6 (left) and see Fig. 7 (left)). $(ii)$ The higher the degree of nonclassicality the quantum state, the higher the phase sensitivity of the interferometer. Thus, the phase sensitivity of two interferometers can be improved by increasing the noncalssicality of input quantum states (see Fig. 6 (right) and see Fig. 7 (right)). $(iii)$ We should emphasize that nonclasscality is not a necessary condition for improving the phase sensitivity of LI. For example, the nonclassicality of $ECS$ and $N00N$ is: $N(\rho _{\text {ecs}})<N(\rho _{N00N})$, but the phase sensitivity for ecs and N00N is:$\ \Delta \phi _{ \text {ecs}}<\Delta \phi _{N00N}$. (see Fig. 6 (right)). In summary, the phase sensitivity of two interferometers is influenced by both entanglement and nonclasscality.

5. Conclusions

We proposed a theoretical model for an interferometer with improved sensitivity, including LI and NI. By extending the method for quantifying nonclassicality of a single-mode quantum state, it was possible to measure the nonclassicality of a two-mode quantum state. Moreover, we discussed the nonclassicality and entanglement properties of several different quantum states. The result show that the N00N is the most nonclassical. In addition, a new normal ordering form of evolution operator of NI was obtained using the techniques of IWOP, and the parity signal of different input state were obtained. By inputting several common quantum states, the phase sensitivity of LI and NI was examined. These results showed that the ecs outperform other quantum states for both LI and NI. Furthermore, we quantitatively investigated the effect of nonclassicality and entanglement on the phase sensitivity of LI and NI. It was determined that nonclassicality or entanglement is a sufficient but necessary condition for improving the phase sensitivity of an interferometer. It should be noted that in this study, external environmental effects, such as the photon loss [45] and phase diffusion [46], were not considered from the perspective of the phase sensitivity of interferometers. However, this would make an interesting topic for future research. In addition to this, the phase sensitivity of interferometers can be considerably improved by adding additional operations, such as quantum feedback [47], adaptive protocols [48] , and the inverse operation scheme [49]. By combining such operations with the entanglement or nonclassicality properties of quantum states, the phase sensitivity of interferometers may be greatly improved. Presently, theoretical research on the improvement of the phase sensitivity of interferometers is relatively mature, and several relevant optical phase measurements have been performed that approach the Heisenberg limit [50]. Moreover, the estimation of multiple phases with limited data has been successfully realized [51].

Our findings are potentially useful for future theoretical and experimental studies in the field of phase estimation theory. By realizing the measurement of the nonclassicality of quantum states [52], it is possible that experiments on the improvement of the phase sensitivity of interferometers via entanglement can be executed in the near future. It is also possible to apply theory to increase the phase sensitivity of analogous interferometers for application to high resolution quantum measurement [53] and the detection of gravitational waves [54].