Abstract

Mach-Zehnder interferometer is a common device in quantum phase estimation and the photon losses in it are an important issue for achieving a high phase accuracy. Here we thoroughly discuss the precision limit of the phase in the Mach-Zehnder interferometer with a coherent state and a superposition of coherent states as input states. By providing a general analytical expression of quantum Fisher information, the phase-matching condition and optimal initial parity are given. Especially, in the photon loss scenario, the sensitivity behaviors are analyzed and specific strategies are provided to restore the phase accuracies for symmetric and asymmetric losses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum metrology, an emerging quantum technology, has been widely studied [1–13] and applied in various sciencfic tasks in recent years, including the detection of gravitational wave [14–16], quantum imaging [17–20] and even biology science [21]. Quantum phase estimation via interferometers is an important topic in quantum metrology. A successful example of phase estimation with interferometers is the Laser Interferometer Gravitational-Wave Observatory (LIGO), which has already catch the signal of gravitational waves [22] in 2015. Another two on-going projects LISA [23] and TianQin [24] are also based on the orbital optical interferometers. Hence, the study of optical phase estimation, especially quantum phase estimation, will definitely promote the technological development in these fields, and may even breed the next-generation detectors for gravitational waves and dark matters.

A su(2) interferometer can be constructed via a Mach-Zehnder interferometer, which typically consists of two beam splitters and one phase shift in one arm, as shown in Fig. 1. Since Caves found the effects of vacuum fluctuation to the phase accuracy in Mach-Zehnder interferometers [25], various types of input states have been discussed, including squeezed state [25–28], NOON state [29–31], entangled coherent state [32–38], BAT state [39], and number squeezed state [40].

 figure: Fig. 1

Fig. 1 Schematic of an Mach-Zehnder Interferometer. The input ports are labeled as A and B. The photon losses in the interferometer are simulated with two fictitious beam splitters, with corresponding operators BACT1 and BBDT2. Here C and D are fictitious ports and T1, T2 are transmission rates. No photon losses exist for T1 = T2 = 1.

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A powerful theoretical tool in quantum parameter estimation to depict the precision limit is the quantum Cramér-Rao bound, which is δϕ^1/μF [41–44]. Here δϕ^ is the standard deviation of parameter ϕ with unbiased estimator ϕ^, µ is the repeated number of experiments and F is the quantum Fisher information (QFI). The most useful resource in the Mach-Zehnder interferometer is the average photon numer n¯, of which the corresponding standard quantum limit for δϕ^ is 1/n¯ and the Heisenberg limit (or Heisenberg scaling) is 1/n¯.

Noise is the major obstacle for obtaining high precision result in quantum parameter estimation. For a large-scale, especially an in-orbit quantum interferometer (in the size of LISA and TianQin), the photon losses between the satellites could be an important issue for a high phase sensitivity. Therefore, fully understanding on the sensitivity behaviors under photon losses in the interferometer could help to restore a high precision as required. Many lossy scenarios with different input states have been discussed in the literature [32, 33, 45–48]. It is common to simulate the photon losses with fictitious beam splitters in theory, as shown in Fig. 1. In this paper, we discuss the precision limit of a Mach-Zehnder interferometer with a coherent state and a superposition of coherent states as the input states. Both perfect and imperfect (with photon losses) scenarios are considered and the analytical expression of QFI is provided. With this expression, the phase-matching condition (PMC) of the input states and the optimal QFI are calculated. For the imperfect scenario, symmetric and asymmetric losses are both studied and corresponding strategies to restore the accuracy are provided.

2. Preliminary knowledge

The quantum Fisher information (QFI) for a parameter ϕ is defined as F := Tr(ρϕL2), where ρϕ is the parameterized state and L is the symmetric logarithmic derivative operator satisfying ϕρϕ=12(Lρϕ+ρϕL). Several methods for the calculation of QFI have been developed in recent years [49–55]. Utilizing the spectral decomposition ρϕ=i=1Mpi|ψiψi|, with M, pi and |ψi〉 the dimension of the support for the matrix, the eigenvalues and eigenstates of ρϕ, the QFI can be expressed by [49–51]:

F=i=1M(ϕpi)2pi+i=1M4piϕψi|ϕψii,j=1M8pipjpi+pj|ψi|ϕψj|2.

For the unitary parameterization process ρϕ = eiHϕ ρeiHϕ, with H a Hermitian operator, the expression of the QFI reduces to

F=i=1M4piψi|H2|ψii,j=1M8pipjpi+pj|ψi|H|ψj|2.

Furthermore, for a pure state |ψϕ〉, it reduces to

F=4(ψϕ|H2|ψϕ|ψϕ|H|ψϕ|2).

In this paper, we focus on the phase estimation in the Mach-Zehnder interferometer, as shown in Fig. 1. The interferometer consists of two beam splitters and a phase shift. The two beam splitters are usually taken as 50:50 beam splitters, which in theory can be expressed by Bx(±π2)=exp(±iπ2JxAB). Here JxAB is a Schwinger operator defined as JxAB=12(ab+ba) with a (b) the annihilation operator for port A (B) and a (b) the corresponding creation operators. The other two Schwinger operators are JyAB=12i(abba) and JzAB=12(aabb) The Schwinger operators satisfy the su(2) algebra. The operator for the phase shift is U(ϕ)=exp(iϕJzAB). For a perfect Mach-Zehnder interferometer, one expression of the entire setup can be written as the unitary operator below [56]

UMZ=Bx(π2)U(ϕ)Bx(π2)=exp(iϕJyAB).

The photon loss in the interferometer is usually depicted via fictitious beam splitters in theory. The effect of a general beam splitter for ports X and Y in quantum optics can be written as BXYT=exp(i2arccosTJxXY) [47,48], where T is the transmission rate. When T = 1 (T = 0), all photons are transmitted (reflected). In many scenarios, especially in the large-scale interferometers, the optical path length is long and the dispersion of light spot is inevitable during the propagation, which will cause photon losses at the second beam splitter. In this paper, ports A and B are input ports of the interferometer and ports C and D are the fictitious ports for photon losses. The effects of fictitious beam splitters are expressed by BACT1 and BBDT2. The arm with respect to port A (B) has no photon losses for T1 = 1 (T2 = 1) and all photons are lost for T1 = 0 (T2 = 0).

3. Perfect interferometer

For the perfect interferometer and with a pure input state, the QFI can be directly obtained by substituting Eq. (4) into Eq. (3), which is

F=(2n¯An¯B+n¯A+n¯B2|a|2|b|2)2Re(a2b2a2b2),
where Re(·) represent the real part and n¯A=aa, n¯B=bb are the average photon numbers for both arms.

Taking a coherent state |β〉 as the input state for port A, and an arbitrary pure state |ψ〉 for port B, n¯A=|a|2=|β|2. The QFI then reads

F=2n¯A(n¯B|b|2)+n¯A+n¯B2Re[β2(b2b2)].

For a given |ψ〉, F only depends on the value of β, and the PMC optimizing the QFI is

|Arg(β2)Arg(b2b2)|=π,

where Arg(·) is the argument. With this condition, the optimal QFI can be calculated as

Fm=2n¯A(n¯B|b|2+|b2b2|)+n¯A+n¯B.

Next we take the input state in port B as the superposition of two coherent states |α〉 and |− α〉, i.e.,

|ψ=Nα(|α+eiΘ|α),
where Θ ∈ [0, 2π) is the relative phase and the normalization factor Nα reads Nα = (2 + 2e−2|α|2 cos Θ)−1/2. In the following we denote β=|β|eiΦA, α=|α|eiΦB with ΦA, ΦB the arguments of β and α. Through some straightforward calculation, the PMC for optimal QFI can be written as
|ΦAΦB|=π2,
which coincides with the case that using a coherent superposition state in port B [57], namely, the relative phase Θ doesn’t affect the PMC. Under this condition, the maximal QFI reads
Fm=n¯A(2|α|2+1)+n¯B(2n¯A+1).

Utilizing the equations n¯A=|β|2 and n¯B=2Nα2|α|2(1e2|α|2cosΘ), the maximal Fm can be reached when Θ = π, which means taking into account the PMC, the QFI can be further improved with an initial odd parity of |ψ〉.

Now we compare Fm with Heisenberg scaling. Denote n¯=n¯A+n¯B as the average total input photon number, Eq. (10) can then be rewritten into Fm=n¯+2n¯A(n¯B+|α|2). For a large |α|, n¯B|α|2, Fmn¯+4n¯An¯B=n¯+n¯2(δn¯)2 with δn¯=n¯An¯B the photon difference between two ports. When δn¯ is small (compared to n¯), Fm reduces to n¯+n¯2, i.e., Fmn¯2, indicating the QFI under PMC can reach the Heisenberg scaling even no initial parity exists in |ψ〉. Furthermore, it can be found that

Fmn^2,
which can be proved as Fmn^2=(|β|2|α|2)20. Here we used n^2=n¯A2+2n¯An¯B+n¯+|α|4. This upper bound can be achieved for |β| = |α|. To satisfy this condition, one can take β=αei(Φ+π2) with Φ the relative phase between the values of α and β, hence the total input state is Nα|αei(Φ+π2)A(|αB+eiΘ|αB). According to Eq. (6), the PMC is Φ = 0 or π, which is indeed independent of Θ.

4. Imperfect interferometer

For an imperfect Mach-Zehnder interferometer, the total effect cannot be treated as an unitary operation. As discussed in the previous section, the photon losses are simulated with beam splitters BACT1=exp[i2arccosT1JxAC] and BBDT2=exp[i2arccosT2JxBD]. Here JxAC=12(ac+ca), JxBD=12(bd+db) with c (c) and d (d) the annihilation (creation) operators of the fictitious lossy ports C and D. We take the total input state as

Nα|αei(Φπ2)A(|αB+eiΘ|αB).

After the photon losses, the state becomes a mixed state, which can be written as (the basis information and detailed calculation can be found in the appendix)

ρ1=Nα2(1+pt2+2e2|α|2cosΘ1pt2(pt+preiΘ)ei|α|2δTsinΦ1pt2(pt+preiΘ)ei|α|2δTsinΦ1pt2),
where pt=e|α|2T, pr=e|α|2R and T = T1 + T2 is the total transmission rate of the photon losses, R = 2 − T is the total reflection rate, δT = T1T2 is the transmission difference between the two ports. Since the last 50:50 beam splitter in the interferometer does not affect the value of QFI as it is independent of θ, the total effect of the lossy interferometer is equivalent to perform the phase shift transform U (θ) to ρ1. Denote the eigenvalues and eigenstates of ρ1 as λ± and |λ±〉, respectively, the QFI can be expressed by
F=i=±4λiλi|(JzAB)2|λii,j=±8λiλjλi+λj|λi|JzAB|λj|2.

Utilizing the expressions of λ± and |λ±〉 (given in the appendix) and through some tedious calculation, the specific expression of QFI can be written as

F=2(δT)2Nα6|α|4ptΔ(1pt2)[4pr(pr+ptcosΘ)(pt+prcosΘ)ΔNα4pt]16(δT)2Nα8Δ|α|4(1pr2)e4|α|2sin2Θ2δTNα2|α|2e2|α|2(4TNα2|α|21)sinΘsinΦ+2T2|α|4Nα2[12Nα2(1pr2)(Δ2Nα2+2Nα2e4|α|2sin2Θ)sin2Φ]+2TNα2|α|2.
where Δ=14detρ1=14Nα4(1pr2)(1pt2). Next we will discuss the PMCs and maximum QFIs for symmetric and asymmetric losses scenarios.

4.1. Symmetric losses

We first consider the symmetric losses case. In this case, the transmission rate in both arms are equivalent, i.e., δT = 0. With this condition, the QFI in Eq. (15) reduces to

F=2T2|α|4Nα2[12Nα2(1pr2)(Δ2Nα2+2Nα2e4|α|2sin2Θ)sin2Φ]+2TNα2|α|2.

To maximize F, the corresponding PMC is Φ = 0 or π, which is the same with lossless case. Utilizing the PMC, Fm is in the form

Fm=2TNα2|α|2{1+T|α|2[12Nα2(1pr2)]}.

Recall the fact that Nα is a function of Θ, Fm can be further improved by optimizing Θ. Utilizing the equation FmNα2=0, it can be found the extremal value of Fm is reached at

Nα2=Nex:=1+T|α|24T|α|2(1pr2)
and due to the fact 2Fm(Nα2)2<0, this extremal value is the maximum value. One may notice that Nα2 is a bounded function with respect to Θ,
Nα2[12(1+e2|α|2),12(1e2|α|2)],
where the lower and upper bounds of Nα2 can be attained at Θ = 0 and π, respectively. To obtain the actual maximum value of Fm, whether Nex locates in above regime needs to be considered. The specific relation between Nex and above regime is shown in Fig. 2. The areas below the solid black line and above the dashed black line represent the regimes that Nex<(2+2e2|α|2)1 and Nex>(22e2|α|2)1, respectively. The area between these lines represents the regime that Nex[(2+2e2|α|2)1,(22e2|α|2)1].

 figure: Fig. 2

Fig. 2 The values of maxΘFm as a function of T and |α|. The areas below the solid black line, above the dashed black line and between these lines represent the regimes that Nex<(2+2e2|α|2)1 (optimal Θ is 0), Nex<(22e2|α|2)1 (optimal Θ is π) and Nex[(2+2e2|α|2)1,(22e2|α|2)1]. The PMC here is Φ = 0, π. maxΘFm takes the logarithmic values in the figure.

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In the regime that Nex>(22e2|α|2)1. Nex is not attainable and the maximum value of Fm with respect to Θ is obtained at Θ = π, namely, an initial odd parity is required. In this case,

maxΘFm=T|α|21e2|α|2[1+T|α|2(11e2|α|2R1e2|α|2)].

Similarly, in the regime that Nex<(2+2e2|α|2)1, Nex is also not attainable and the maximum value of Fm with respect to Θ is obtained at Θ = 0, namely, an initial even parity is required for optimal Fm. In this case,

maxΘFm=T|α|21+e2|α|2[1+T|α|2(11e2|α|2R1+e2|α|2)].

In the regime that Nex[(2+2e2|α|2)1,(22e2|α|2)1], Nex is reachable and the maximum Fm can be attained at Nα2=Nex. The maximum Fm reads

maxΘFm=(1+T|α|2)24(1e2|α|2R).

The optimal Θ satisfies the following equation

cosΘ=2T|α|2e2|α|2(1T)+(T|α|21)e2|α|21+T|α|2.

In this regime, the optimal Θ relies on the values of T, |α| and both odd and even input states are non-optimal.

From the lines shown in Fig. 2, it can be seen the area between the lines is small, which means for most values of T and |α|, Nex is out of the regime [(2+2e2|α|2)1,(22e2|α|2)1]

, and initial parity will benefit the precision limit. Besides, though the PMC here is not changed compared to the lossless scenario, the maximum Fm with respect to Θ is different for different parameter regimes as discussed above. However, in all regimes, increasing the intensity of initial state always benefits the precision limit, as shown in Fig. 2. Thus, for an intermediate photon loss rate, the best strategy to hold the precision limit is to use a high intensity odd state as the input state. However, for a low photon loss rate, one should be more careful since the increasing of the intensity may requires a changed parity for optimal precision limit. And to keep the odd parity to be optimal, a higher intensity is required with the decrease of the photon loss rate (the increase of the transmission rate T).

4.2. Asymmetric losses

For asymmetric losses scenario, δT ≠ 0. To find the PMC, the derivative of QFI on sin Φ needs to be calculated. Based on Eq. (15), it is

FsinΦ=2δTNα2|α|2e2|α|2sinΘ(14TNα2|α|2)2T2|α|4(Δ+4Nα4e4|α|2sin2Θ)sinΦ.

Due to the fact 2F(sinΦ)2<0, the solution for above equation gives the maximum value of QFI, i.e., the optimal Φ needs to satisfy

sinΦ=Nex:=Nα2(4TNα2|α|21)sinΘT2|α|2(Δe2|α|2+4Nα4e2|α|2sin2Θ)δT.

The solution for this equation relies on the values of Θ. However, similar to the symmetric scenario, sin Φ is restrained in the regime [−1, 1]. Hence, when Nex[1,1] the PMC is the solution for above equation, especially, if the input state is an even state, i.e., Θ = 0, the PMC then reads Φ = 0. For case that Nex>1, the PMC is Φ = π/2, and for Nex<1, the PMC is Φ = 3π/2.

For a low intensity input (|α| is very small), Nex12TtanΘ2sinΘδT4T2|α|2. Its relation between the regime [−1, 1] highly relies on the value of Θ and the sign of δT, indicating no constant PMC exists. A more concerned case is with a high intensity input. In this case, Nα21/2 and Δe2|α|2e2|α|2(1T)+e2|α|2(1R). Since 1 − T and 1 − R always take different signs as T + R = 2, Δe2|α|2 is very large here. Thus, Nex approximates to zero. Based on Eq. (25), the PMC here is Φ ≈ 0 or π. Namely, for an asymmetric loss scenario with a high enough intensity input, the PMC can still be independent of Θ and the transmission rates. Another benefit of this PMC is that Fm is insensitive to the sign of δT (since it only exists together with sin Φ), i.e., the information of which path has a more severe leak is not required here. Taking this PMC (Φ = 0, π), Fm approximates to

FmT|α|2,
which is independent of Θ. Figure 3 shows the difference between Eq. (15) and T|α|2 for different parameter settings. Generally, a larger transmission rate requires a larger |α| to converge to T|α|2. However, for the specific parameters given in the figure, all Fm converge before around |α| = 3.0, which means T|α|2 is a very good approximation here for |α| > 3.0 regardless the values of T, δT and Θ. Thus, for the asymmetric losses scenario, one efficient strategy to restore the precision limit is inputting a high intensity state satisfying the PMC.

 figure: Fig. 3

Fig. 3 The difference between Eq. (15) and T |α|2 under the PMC Φ = 0. The expression T |α|2 is a good approximation from |α| ≈ 3 for the coefficients values in the figure. A larger transmission rate T requires a larger |α| for this approximation.

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4.3. The scaling

The average total photon number n¯ for the input state in Eq. (12) is

n¯=2|α|21+e2|α|2cosΘ.

For a large |α|, n¯2|α|2. In the symmetric losses scenario, Eqs. (20) and (21) indicate maxΘFmn¯ for a nonzero R, which means it can only reach the standard quantum limit in these regime. For Eq. (22), maxΘFmn¯2, i.e., it can still attain the Heisenberg scaling, however, the allowed value Θ of |α| in this regime (shown in Fig. 2) is very limited, and the absolute value of maxΘFm is still worse than Eq. (20) with a large |α|. Therefore, the scaling of maxΘFm for symmetric losses can only provide a precision at the standard quantum limit. For asymmetric losses scenario, taking into account the PMC, Fm approximates to T |α|2 for high intensity state, i.e., proportional to n¯, the standard quantum limit. This phenomenon coincides with some other cases that the precision limit is bounded by the standard quantum limit when local noise exists [3, 4, 49, 58, 59].

5. Conclusion

This paper focuses on the phase estimation of a su(2) Mach-Zehnder interferometer, in which the unknown parameter ϕ is encoded by a phase shift in one arm. The input states is a coherent state |β〉 and a superposition state of coherent states Nα(|α〉 + eiΘ| −α〉). Both perfect and imperfect scenarios are considered. For the perfect scenario, the phase-matching condition to optimize the QFI is given. With this condition, the QFI can be further improved by taking Θ = π, i.e., using an odd parity state (cat state). For the imperfect scenario, the photon losses in both arms are simulated by two fictitious beam splitters. The general analytical expression of QFI are provided, as well as the phase-matching condition and optimal Θ to maximize QFI. In the symmetric losses case, the phase-matching condition is unchanged compared to the lossless case. Furthermore, there exists a small parameter regime for total transmission rate T and |α| that optimal Θ is sensitive to them. To avoid this regime, one strategy is using a high intensity input state, of which the precision limit is at the standard quantum limit. However, it should be noticed that for a large T, increasing the intensity may requires the change of parity from even to odd in the mean time, and to keep the odd parity as the optimal one, a higher intensity is required with the decrease of the photon loss rate (increase of T). In the asymmetric losses case, taking the approximated phase-matching condition, an efficient strategy to avoid the sensitivity of maximum QFI on Θ and restore the sensitivity is also using the a high intensity input state satisfying the PMC.

Appendix Derivation of QFI for imperfect interferometer

The input state we choose in this paper is

|ψin=|αei(Φ+π2)Nα(|α+eiΘ|α).

For the first 50:50 beam splitter, the state becomes |ψ0=Bx(π2)|ψin. Utilizing the formula

BxT|αA|βA=|αT+iβ1TA|βT+iα1TA,
where BxT=exp(i2arccosTJxAA) and being aware of the fact T = 1/2 for 50:50 beam splitter, |ψ0〉 can be written as
|ψ0=Nα(|if+A|fB+eiΘ|ifA|f+B),
where f±:=α2(1±eiΦ). Recall the fictitious beam splitters BACT1, BBDT1 as exp(i2arccosT1JxAC) and exp(i2arccosT2JxBD), where C, D are labels of two fictitious output ports with c(c) and d (d) the annihilation (creation) operators and JxAC=12(ac+ac), JxBD=12(bd+bd). Assume the input states of the fictitious input ports are vacuum, and after the photon losses, the output state |ψ1〉 can be written as
|ψ1=Nα(|A|f+R1C|ifR2D+eiΘ||fR1C|if+R2D),
where |A:=|if+T1A|fT2B and |:=|ifT1A|f+T2B. R1 = 1 −T1, R2 = 1 −T2 are the reflection rates. The reduced matrix can then be calculated as
ρ1=TrCD(|ψ1ψ1|)=Nα2[|AA|+||+prei(|α|2δTsinΦΘ)|A|+prei(|α|2δTsinΦΘ)|A|],
where pr=e|α|2R with δT = T1T2 the transmission difference. Notice |A and | are not orthogonal due to the fact A|=ptei|α|2δTsinΦ with pt=e|α|2T and T = T1 + T2 the total transmission rate.

Now we introduce an orthogonal basis {|A,|A}, where

|A=11pt2(|ptei|α|2δTsinΦ|A).

In this basis, ρ1 can be written as

ρ1=Nα2(1+pt2+2e2|α|2cosΘ1pt2(pt+preiΘ)ei|α|2δTsinΦ1pt2(pt+preiΘ)ei|α|2δTsinΦ1pt2),

The eigenvalues for this matrix are λ±=Nα2(1±Δ)/2 and corresponding eigenstates |λ±〉 are

|λ±=(v±pt+preiΘpt2+pr2+2e2|α|2cosΘptv1pt2)|A±vei|α|2δTsinΦ1pt2|,
where we have used the expression of |A. The coefficients read
v±=121±pt2+e2|α|2cosΘΔ(1+e2|α|2cosΘ),
=1ΔΔ2±ΔNα2(pt2+e2|α|2cosΘ),
Δ=14detρ1.

Funding

National Key Research and Development Program of China (2017YFA0205700 and 2017YFA0304202); National Natural Science Foundation of China (NSFC) (11475146); Fundamental Research Funds for the Central Universities (2017FZA3005).

Acknowledgments

The authors thank X. Xiao and J. Chen for helpful discussions.

Disclosures

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

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23. S. Vitale, “Multiband gravitational-wave astronomy: parameter estimation and tests of general relativity with space-and ground-based detectors,” Phys. Rev. Lett. 117, 051102 (2016). [CrossRef]  

24. J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016). [CrossRef]  

25. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981). [CrossRef]  

26. L. Pezze and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008). [CrossRef]   [PubMed]  

27. P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95, 023824 (2017). [CrossRef]  

28. P. Liu and G. R. Jin, “Ultimate phase estimation in a squeezed-state interferometer using photon counters with a finite number resolution,” J. Phys. A: Math. Theor. 50, 405303 (2017). [CrossRef]  

29. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994). [CrossRef]   [PubMed]  

30. D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004). [CrossRef]   [PubMed]  

31. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013). [CrossRef]   [PubMed]  

32. J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011). [CrossRef]   [PubMed]  

33. J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86, 043828 (2012). [CrossRef]  

34. J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A: Math. Theor. 49, 115302 (2016). [CrossRef]  

35. Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.

36. B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811 (1992). [CrossRef]   [PubMed]  

37. C. C. Gerry, “Generation of Schrödinger cats and entangled coherent states in the motion of a trapped ion by a dispersive interaction,” Phys. Rev. A 55, 2478 (1997). [CrossRef]  

38. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010). [CrossRef]  

39. J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010). [CrossRef]  

40. L. Pezze and A. Smerzi, “Ultrasensitive two-mode interferometry with single-mode number squeezing,” Phys. Rev. Lett. 110, 163604 (2013). [CrossRef]   [PubMed]  

41. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

42. A. S. Holevo, Probabilistic and Statistic Aspects of Quantum Theory (North-Holland, 1982).

43. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994). [CrossRef]   [PubMed]  

44. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef]   [PubMed]  

45. P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014). [CrossRef]  

46. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017). [CrossRef]  

47. R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009). [CrossRef]  

48. N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012). [CrossRef]  

49. Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013). [CrossRef]  

50. J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014). [CrossRef]  

51. J. Liu, H.-N. Xiong, F. Song, and X. Wang, “Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks,” Physica A 410, 167–173 (2014). [CrossRef]  

52. S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014). [CrossRef]  

53. Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014). [CrossRef]  

54. J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015). [CrossRef]   [PubMed]  

55. J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016). [CrossRef]  

56. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986). [CrossRef]  

57. J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013). [CrossRef]  

58. M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015). [CrossRef]  

59. R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics , 60, 345–435 (2015). [CrossRef]  

References

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  9. H. Yuan and C.-H. F. Fung, “Quantum parameter estimation with general dynamics,” Npj:Quantum Information 3, 14 (2017).
  10. Y. Yao, L. Ge, X. Xiao, X. Wang, and C. P. Sun, “Multiple phase estimation for arbitrary pure states under white noise,” Phys. Rev. A 90, 062113 (2014).
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  29. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
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  30. D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
    [Crossref] [PubMed]
  31. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
    [Crossref] [PubMed]
  32. J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
    [Crossref] [PubMed]
  33. J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86, 043828 (2012).
    [Crossref]
  34. J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A: Math. Theor. 49, 115302 (2016).
    [Crossref]
  35. Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.
  36. B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811 (1992).
    [Crossref] [PubMed]
  37. C. C. Gerry, “Generation of Schrödinger cats and entangled coherent states in the motion of a trapped ion by a dispersive interaction,” Phys. Rev. A 55, 2478 (1997).
    [Crossref]
  38. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
    [Crossref]
  39. J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010).
    [Crossref]
  40. L. Pezze and A. Smerzi, “Ultrasensitive two-mode interferometry with single-mode number squeezing,” Phys. Rev. Lett. 110, 163604 (2013).
    [Crossref] [PubMed]
  41. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  42. A. S. Holevo, Probabilistic and Statistic Aspects of Quantum Theory (North-Holland, 1982).
  43. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
    [Crossref] [PubMed]
  44. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [Crossref] [PubMed]
  45. P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
    [Crossref]
  46. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017).
    [Crossref]
  47. R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
    [Crossref]
  48. N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
    [Crossref]
  49. Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
    [Crossref]
  50. J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014).
    [Crossref]
  51. J. Liu, H.-N. Xiong, F. Song, and X. Wang, “Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks,” Physica A 410, 167–173 (2014).
    [Crossref]
  52. S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014).
    [Crossref]
  53. Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
    [Crossref]
  54. J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015).
    [Crossref] [PubMed]
  55. J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
    [Crossref]
  56. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
    [Crossref]
  57. J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013).
    [Crossref]
  58. M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
    [Crossref]
  59. R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
    [Crossref]

2017 (6)

J. Liu and H. Yuan, “Quantum parameter estimation with optimal control,” Phys. Rev. A 96, 012117 (2017).
[Crossref]

J. Liu and H. Yuan, “Control-enhanced multiparameter quantum estimation,” Phys. Rev. A 96, 042114 (2017).
[Crossref]

H. Yuan and C.-H. F. Fung, “Quantum parameter estimation with general dynamics,” Npj:Quantum Information 3, 14 (2017).

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95, 023824 (2017).
[Crossref]

P. Liu and G. R. Jin, “Ultimate phase estimation in a squeezed-state interferometer using photon counters with a finite number resolution,” J. Phys. A: Math. Theor. 50, 405303 (2017).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017).
[Crossref]

2016 (7)

J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
[Crossref]

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

B. P. Abbott and LIGO Scientific Collaboration and the Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

S. Vitale, “Multiband gravitational-wave astronomy: parameter estimation and tests of general relativity with space-and ground-based detectors,” Phys. Rev. Lett. 117, 051102 (2016).
[Crossref]

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
[Crossref]

J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A: Math. Theor. 49, 115302 (2016).
[Crossref]

H. Yuan, “Sequential feedback scheme outperforms the parallel scheme for Hamiltonian parameter estimation,” Phys. Rev. Lett. 117, 160801 (2016).
[Crossref] [PubMed]

2015 (4)

H. Yuan and C.-H. F. Fung, “Optimal feedback scheme and universal time scaling for Hamiltonian parameter estimation,” Phys. Rev. Lett. 115, 110401 (2015).
[Crossref] [PubMed]

J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015).
[Crossref] [PubMed]

M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
[Crossref]

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
[Crossref]

2014 (8)

J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014).
[Crossref]

J. Liu, H.-N. Xiong, F. Song, and X. Wang, “Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks,” Physica A 410, 167–173 (2014).
[Crossref]

S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014).
[Crossref]

Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
[Crossref]

P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
[Crossref]

R. Demkowicz-Dobrzanski and L. Maccone, “Using entanglement against noise in quantum metrology,” Phys. Rev. Lett. 113, 250801 (2014).
[Crossref]

Y. Yao, L. Ge, X. Xiao, X. Wang, and C. P. Sun, “Multiple phase estimation for arbitrary pure states under white noise,” Phys. Rev. A 90, 062113 (2014).
[Crossref]

G. Toth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

2013 (5)

M. Tsang, “Quantum metrology with open dynamical systems,” New J. Phys. 15, 073005 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

L. Pezze and A. Smerzi, “Ultrasensitive two-mode interferometry with single-mode number squeezing,” Phys. Rev. Lett. 110, 163604 (2013).
[Crossref] [PubMed]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013).
[Crossref]

2012 (4)

N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
[Crossref]

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86, 043828 (2012).
[Crossref]

R. W. Boyd and J. P. Dowling, “Quantum lithography: Status of the field,” Quantum Inf. Process. 11, 891 (2012).
[Crossref]

R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

2011 (3)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
[Crossref] [PubMed]

2010 (3)

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
[Crossref]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010).
[Crossref]

2009 (2)

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inform. 7, 125 (2009).
[Crossref]

2008 (3)

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[Crossref]

L. Pezze and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

2007 (1)

Y. H. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016 (2007).
[Crossref]

2006 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

2004 (1)

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

1999 (1)

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature 400, 849–852 (1999).
[Crossref]

1998 (1)

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
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1997 (1)

C. C. Gerry, “Generation of Schrödinger cats and entangled coherent states in the motion of a trapped ion by a dispersive interaction,” Phys. Rev. A 55, 2478 (1997).
[Crossref]

1994 (2)

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
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S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
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1992 (1)

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811 (1992).
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1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
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1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
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Abbott, B. P.

B. P. Abbott and LIGO Scientific Collaboration and the Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
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Apellaniz, I.

G. Toth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
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Banaszek, K.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Barbieri, M.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Barrett, M. D.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
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Bollinger, J. J.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
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Bouyer, P.

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
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Bowen, W. P.

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
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Boyd, R. W.

R. W. Boyd and J. P. Dowling, “Quantum lithography: Status of the field,” Quantum Inf. Process. 11, 891 (2012).
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Braunstein, S. L.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

Britton, J.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
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Brun, T. A.

S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014).
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Cable, H.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
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Caves, C. M.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Chen, J.

J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
[Crossref]

Chen, L.-S.

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
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Chiaverini, J.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Chu, S.

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature 400, 849–852 (1999).
[Crossref]

Chung, K. Y.

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature 400, 849–852 (1999).
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Cohen, L.

Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.

Cooper, J. J.

N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
[Crossref]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010).
[Crossref]

Datta, A.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Davidovich, L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

de Matos Filho, R. L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
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Demkowicz-Dobrzanski, R.

M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
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R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
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R. Demkowicz-Dobrzanski and L. Maccone, “Using entanglement against noise in quantum metrology,” Phys. Rev. Lett. 113, 250801 (2014).
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R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Dorner, U.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

Dowling, J. P.

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017).
[Crossref]

R. W. Boyd and J. P. Dowling, “Quantum lithography: Status of the field,” Quantum Inf. Process. 11, 891 (2012).
[Crossref]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Duan, H.-Z.

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
[Crossref]

Dunningham, J. A.

P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
[Crossref]

N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
[Crossref]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010).
[Crossref]

Eisenberg, H. S.

Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.

Escher, B. M.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

Fung, C.-H. F.

H. Yuan and C.-H. F. Fung, “Quantum parameter estimation with general dynamics,” Npj:Quantum Information 3, 14 (2017).

H. Yuan and C.-H. F. Fung, “Optimal feedback scheme and universal time scaling for Hamiltonian parameter estimation,” Phys. Rev. Lett. 115, 110401 (2015).
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Ge, L.

Y. Yao, L. Ge, X. Xiao, X. Wang, and C. P. Sun, “Multiple phase estimation for arbitrary pure states under white noise,” Phys. Rev. A 90, 062113 (2014).
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Gerry, C. C.

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
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C. C. Gerry, “Generation of Schrödinger cats and entangled coherent states in the motion of a trapped ion by a dispersive interaction,” Phys. Rev. A 55, 2478 (1997).
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Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
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Gkortsilas, N.

N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
[Crossref]

Glasser, R. T.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[Crossref]

Gong, Y.-G.

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
[Crossref]

Guta, M.

R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Hallwood, D. W.

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81, 043624 (2010).
[Crossref]

Haritos, K. G.

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
[Crossref]

Heinzen, D. J.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
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C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Holevo, A. S.

A. S. Holevo, Probabilistic and Statistic Aspects of Quantum Theory (North-Holland, 1982).

Hu, S.

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
[Crossref]

Humphreys, P. C.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Huver, S. D.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Israel, Y.

Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.

Itano, W. M.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
[Crossref] [PubMed]

Izumi, S.

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017).
[Crossref]

Jarzyna, M.

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
[Crossref]

M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
[Crossref]

Jeong, H.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86, 043828 (2012).
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Ji, J.

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
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Jiang, Z.

Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
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Jin, G. R.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95, 023824 (2017).
[Crossref]

P. Liu and G. R. Jin, “Ultimate phase estimation in a squeezed-state interferometer using photon counters with a finite number resolution,” J. Phys. A: Math. Theor. 50, 405303 (2017).
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Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
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Jing, X.

J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013).
[Crossref]

Jing, X.-X.

J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
[Crossref]

J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015).
[Crossref] [PubMed]

Joo, J.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86, 043828 (2012).
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J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
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Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states by mixing squeezed vacuum and coherent light,” https://arxiv.org/abs/1707.01809.

Jost, J. D.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Kasevich, M. A.

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
[Crossref]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

Knott, P. A.

P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
[Crossref]

Kolodynski, J.

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
[Crossref]

R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Lam, P. K.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
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Langer, C.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Leibfried, D.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Li, X. W.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
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Liu, J.

J. Liu and H. Yuan, “Control-enhanced multiparameter quantum estimation,” Phys. Rev. A 96, 042114 (2017).
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J. Liu and H. Yuan, “Quantum parameter estimation with optimal control,” Phys. Rev. A 96, 012117 (2017).
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J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A: Math. Theor. 49, 115302 (2016).
[Crossref]

J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
[Crossref]

J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015).
[Crossref] [PubMed]

J. Liu, H.-N. Xiong, F. Song, and X. Wang, “Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks,” Physica A 410, 167–173 (2014).
[Crossref]

J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013).
[Crossref]

Liu, P.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95, 023824 (2017).
[Crossref]

P. Liu and G. R. Jin, “Ultimate phase estimation in a squeezed-state interferometer using photon counters with a finite number resolution,” J. Phys. A: Math. Theor. 50, 405303 (2017).
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Figures (3)

Fig. 1
Fig. 1 Schematic of an Mach-Zehnder Interferometer. The input ports are labeled as A and B. The photon losses in the interferometer are simulated with two fictitious beam splitters, with corresponding operators B AC T 1 and B BD T 2. Here C and D are fictitious ports and T1, T2 are transmission rates. No photon losses exist for T1 = T2 = 1.
Fig. 2
Fig. 2 The values of max Θ F m as a function of T and |α|. The areas below the solid black line, above the dashed black line and between these lines represent the regimes that N ex < ( 2 + 2 e 2 | α | 2 ) 1 (optimal Θ is 0), N ex < ( 2 2 e 2 | α | 2 ) 1 (optimal Θ is π) and N ex [ ( 2 + 2 e 2 | α | 2 ) 1 , ( 2 2 e 2 | α | 2 ) 1 ]. The PMC here is Φ = 0, π. max Θ F m takes the logarithmic values in the figure.
Fig. 3
Fig. 3 The difference between Eq. (15) and T |α|2 under the PMC Φ = 0. The expression T |α|2 is a good approximation from |α| ≈ 3 for the coefficients values in the figure. A larger transmission rate T requires a larger |α| for this approximation.

Equations (39)

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F = i = 1 M ( ϕ p i ) 2 p i + i = 1 M 4 p i ϕ ψ i | ϕ ψ i i , j = 1 M 8 p i p j p i + p j | ψ i | ϕ ψ j | 2 .
F = i = 1 M 4 p i ψ i | H 2 | ψ i i , j = 1 M 8 p i p j p i + p j | ψ i | H | ψ j | 2 .
F = 4 ( ψ ϕ | H 2 | ψ ϕ | ψ ϕ | H | ψ ϕ | 2 ) .
U MZ = B x ( π 2 ) U ( ϕ ) B x ( π 2 ) = exp ( i ϕ J y AB ) .
F = ( 2 n ¯ A n ¯ B + n ¯ A + n ¯ B 2 | a | 2 | b | 2 ) 2 Re ( a 2 b 2 a 2 b 2 ) ,
F = 2 n ¯ A ( n ¯ B | b | 2 ) + n ¯ A + n ¯ B 2 Re [ β 2 ( b 2 b 2 ) ] .
| Arg ( β 2 ) Arg ( b 2 b 2 ) | = π ,
F m = 2 n ¯ A ( n ¯ B | b | 2 + | b 2 b 2 | ) + n ¯ A + n ¯ B .
| ψ = N α ( | α + e i Θ | α ) ,
| Φ A Φ B | = π 2 ,
F m = n ¯ A ( 2 | α | 2 + 1 ) + n ¯ B ( 2 n ¯ A + 1 ) .
F m n ^ 2 ,
N α | α e i ( Φ π 2 ) A ( | α B + e i Θ | α B ) .
ρ 1 = N α 2 ( 1 + p t 2 + 2 e 2 | α | 2 cos Θ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ) ,
F = i = ± 4 λ i λ i | ( J z AB ) 2 | λ i i , j = ± 8 λ i λ j λ i + λ j | λ i | J z AB | λ j | 2 .
F = 2 ( δ T ) 2 N α 6 | α | 4 p t Δ ( 1 p t 2 ) [ 4 p r ( p r + p t cos Θ ) ( p t + p r cos Θ ) Δ N α 4 p t ] 16 ( δ T ) 2 N α 8 Δ | α | 4 ( 1 p r 2 ) e 4 | α | 2 sin 2 Θ 2 δ T N α 2 | α | 2 e 2 | α | 2 ( 4 T N α 2 | α | 2 1 ) sin Θ sin Φ + 2 T 2 | α | 4 N α 2 [ 1 2 N α 2 ( 1 p r 2 ) ( Δ 2 N α 2 + 2 N α 2 e 4 | α | 2 sin 2 Θ ) sin 2 Φ ] + 2 T N α 2 | α | 2 .
F = 2 T 2 | α | 4 N α 2 [ 1 2 N α 2 ( 1 p r 2 ) ( Δ 2 N α 2 + 2 N α 2 e 4 | α | 2 sin 2 Θ ) sin 2 Φ ] + 2 T N α 2 | α | 2 .
F m = 2 T N α 2 | α | 2 { 1 + T | α | 2 [ 1 2 N α 2 ( 1 p r 2 ) ] } .
N α 2 = N ex : = 1 + T | α | 2 4 T | α | 2 ( 1 p r 2 )
N α 2 [ 1 2 ( 1 + e 2 | α | 2 ) , 1 2 ( 1 e 2 | α | 2 ) ] ,
max Θ F m = T | α | 2 1 e 2 | α | 2 [ 1 + T | α | 2 ( 1 1 e 2 | α | 2 R 1 e 2 | α | 2 ) ] .
max Θ F m = T | α | 2 1 + e 2 | α | 2 [ 1 + T | α | 2 ( 1 1 e 2 | α | 2 R 1 + e 2 | α | 2 ) ] .
max Θ F m = ( 1 + T | α | 2 ) 2 4 ( 1 e 2 | α | 2 R ) .
cos Θ = 2 T | α | 2 e 2 | α | 2 ( 1 T ) + ( T | α | 2 1 ) e 2 | α | 2 1 + T | α | 2 .
F sin Φ = 2 δ T N α 2 | α | 2 e 2 | α | 2 sin Θ ( 1 4 T N α 2 | α | 2 ) 2 T 2 | α | 4 ( Δ + 4 N α 4 e 4 | α | 2 sin 2 Θ ) sin Φ .
sin Φ = N ex : = N α 2 ( 4 T N α 2 | α | 2 1 ) sin Θ T 2 | α | 2 ( Δ e 2 | α | 2 + 4 N α 4 e 2 | α | 2 sin 2 Θ ) δ T .
F m T | α | 2 ,
n ¯ = 2 | α | 2 1 + e 2 | α | 2 cos Θ .
| ψ in = | α e i ( Φ + π 2 ) N α ( | α + e i Θ | α ) .
B x T | α A | β A = | α T + i β 1 T A | β T + i α 1 T A ,
| ψ 0 = N α ( | i f + A | f B + e i Θ | i f A | f + B ) ,
| ψ 1 = N α ( | A | f + R 1 C | i f R 2 D + e i Θ | | f R 1 C | i f + R 2 D ) ,
ρ 1 = Tr CD ( | ψ 1 ψ 1 | ) = N α 2 [ | A A | + | | + p r e i ( | α | 2 δ T sin Φ Θ ) | A | + p r e i ( | α | 2 δ T sin Φ Θ ) | A | ] ,
| A = 1 1 p t 2 ( | p t e i | α | 2 δ T sin Φ | A ) .
ρ 1 = N α 2 ( 1 + p t 2 + 2 e 2 | α | 2 cos Θ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ) ,
| λ ± = ( v ± p t + p r e i Θ p t 2 + p r 2 + 2 e 2 | α | 2 cos Θ p t v 1 p t 2 ) | A ± v e i | α | 2 δ T sin Φ 1 p t 2 | ,
v ± = 1 2 1 ± p t 2 + e 2 | α | 2 cos Θ Δ ( 1 + e 2 | α | 2 cos Θ ) ,
= 1 Δ Δ 2 ± Δ N α 2 ( p t 2 + e 2 | α | 2 cos Θ ) ,
Δ = 1 4 det ρ 1 .

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