# Improvement of phase sensitivity in SU(1,1) interferometer via a Kerr
nonlinear

Shoukang Chang1, Wei Ye1,4, Huan Zhang1, Liyun Hu1,∗, Jiehui
Huang${}^{2}{\small\thanks{Corresponding author. Email:
jiehuihuang@126.com}},$and Sanqiu Liu1,3 Corresponding author. Email:
<EMAIL_ADDRESS>author. Email<EMAIL_ADDRESS>author. Email<EMAIL_ADDRESS>1Center for Quantum Science and Technology,
Jiangxi Normal University, Nanchang 330022, China 2School of Mathematics,
Physics and Statistics, Shanghai University of Engineering Science, Shanghai
201620, China 3Department of physics, Nanchang University, 330031, China
4School of Computer Science and Engineering, Central South University,
Changsha 410083, China

###### Abstract

We propose a theoretical scheme to enhance the phase sensitivity by
introducing a Kerr nonlinear phase shift into the traditional SU(1,1)
interferometer with a coherent state input and homodyne detection. We
investigate the realistic effects of photon losses on phase sensitivity and
quantum Fisher information. The results show that compared with the linear
phase shift in SU(1,1) interferometer, the Kerr nonlinear case can not only
enhance the phase sensitivity and quantum Fisher information, but also
significantly suppress the photon losses. We also observe that at the same
accessible parameters, internal losses have a greater influence on the phase
sensitivity than the external ones. It is interesting that, our scheme shows
an obvious advantage of low-cost input resources to obtain higher phase
sensitivity and larger quantum Fisher information due to the introduction of
nonlinear phase element.

PACS: 03.67.-a, 05.30.-d, 42.50,Dv, 03.65.Wj

##  I Introduction

Quantum metrology has a close relation to various-important information areas,
such as Bose-Einstein condensate 1 ; 2 ; 3 , gravitational wave detection 4 ;
5 , and quantum imaging 6 ; 7 ; 8 . It has been widely concerned and highly
developed in recent years. To meet the high precision demand, all kinds of
optical interferometers have been proposed. For instance, as a general model,
the Mach–Zehnder interferometer (MZI) has been used to be an essential tool to
provide insight into tiny variations on phase shift 9 ; 10 ; 11 .

In order to improve the precise measurement, generally speaking, we can focus
on the following three stages 12 : probe generation 13 ; 14 ; 15 ; 16 , probe
modification 17 ; 18 ; 19 ; 20 ; 21  and probe readout 22 ; 23 , as
illustrated in Fig. 1(a). In particular, for the probe generation, the phase
sensitivity is always confined to the standard quantum limit (SQL) when the
classical resources are injected into the input ports of the MZI. To surpass
the SQL, non-classical quantum states have widely used as the input of the
MZI, such as entangled states 13 , twin Fock states 24 , and NOON states 25 ,
by which the Heisenberg limit (HL) even can be reached 26 ; 27 . Although the
usage of nonclassical states can greatly improve the phase sensitivity of
optical interferometers, these states with large average photon numbers are
not only more difficult to prepare, but also very fragile especially in the
presence of environmental interferences 28 ; 29 ; 30 . Thus, from the
viewpoint of resource theory, it will be a challenging task with simple input
state (such as coherent state (CS)) to further improve the precision of
measurement, especially in the realistic case.

On the other hand, many efforts have been paid to the stage of probe
modification, especially when Yurke _et al_. first proposed an SU(1,1)
interferometer with a linear phase shift 20 . In this system, the active
nonlinear optical devices, such as four-wave mixers (FWMs) and optical
parametric amplifiers (OPAs), are used instead of the passive linear beam
splitters (BSs) used in the conventional MZI 31 ; 32 ; 33 ; 34 ; 35 ; 36 ; 37
. To beat the SQL, Plick _et al_. applied strong CS as the inputs into the
SU(1,1) interferometer 33 . Subsequently, Li _et al_. proposed a scheme of
reaching HL sensitivity via a squeezed vacuum state (SVS) plus a CS with the
homodyne detection 31 . It is interesting that, Hudelist _et al_. pointed out
that the signal-to-noise ratio (SNR) of the SU(1,1) interferometer is about
4.1 dB higher than that of the MZI under the same phase-sensing intensity 38 .
This point may be one of reasons focusing on this interferometer. Actually,
this implies the role of nonlinear process for improving the precision.

Expect the SU(1,1) nonlinear process, the nonlinear phase shifts have also
been proposed for enhancing the phase estimation, which can be viewed as
another way of the probe modification 9 ; 39 ; 40 ; 41 ; 42 . For instance, in
the traditional MZI, Zhang et al. investigated the phase estimation by
replacing linear phase shift by nonlinear one 9  using a CS and parity
measurement. Jiao et al. proposed an improved protocol of nonlinear phase
estimation by inserting a nonlinear phase shift into the traditional MZI, with
active correlation output readout and homodyne detection 39 . More recently,
Chang et al. suggested a scheme for enhancing phase sensitivity by introducing
nonlinear phase shifter to the modified interferometer consisting of a
balanced BS and an optical parameter amplifier (OPA) 41 . It is shown that the
OPA potential can be stimulated by nonlinear phase shifter, which is absent
for the case of linear phase shifter. In addition, the estimation of nonlinear
phase has also lots of applications, such as the third-order susceptibility of
the Kerr medium 39 , phase-sensitive amplifiers 43 ; 44  and nonclassical
quantum state preparations 45 ; 46 . These results show that the nonlinear
optical devices can be considered as powerful tools to effectively achieve
both high accuracy and sensitivity. However, on one hand, the researches on
nonlinear phase estimation are not as systematic as those on linear one. On
the other, most works on the phase precision are based on either specific
measurement, especially in the presence of photon loss, or direct calculation
of quantum Fisher information for ideal case.

In this paper, we mainly focus on the nonlinear phase estimation of a Kerr
SU(1,1) (KSU(1,1)) interferometer by replacing the linear phase shift with the
Kerr nonlinear one, together with CS plus vacuum state (VS) (denoted as
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$) as inputs and
homodyne detection. Both the phase sensitivity and the quantum Fisher
information (QFI) are analytically investigated with and without photon
losses. The phase sensitivity can beat the SQL and approach the HL. Compared
to the traditional SU(1,1) interferometer with a linear phase and other inputs
resources including CS+CS and SVS+CS, our scheme presents much QFI and higher
phase sensitivity closer to the quantum Cramér-Rao (QCRB) 34 ; 47 ; 48 ; 49 ;
50 . From the viewpoint of resource theory, CS+VS can be seen as the most
simple and easily available input, thus our scheme has an obvious advantage of
low-cost input by inserting nonlinear phase shift into the SU(1,1)
interferometer.

The remainder of this paper is arranged as follows. Section II introduces the
KSU(1,1) interferometer in our scheme. In Section III, we investigate the
phase sensitivity of the output signal with homodyne detection, and compare
them with the conventional SU(1,1) interferometer. In Section IV, we get the
QFI of Kerr nonlinear phase shift for the KSU(1,1) interferometer by invoking
the characteristic function (CF) approach. In Section V, we discuss the
effects of photon losses on both phase sensitivity and QFI of Kerr nonlinear
phase shift, respectively. Conclusions are made in the last section.

Figure 1: (Color online) (a) The general process of estimating an unknown
parameter $\phi$. (b) Schematic diagram of SU(1,1) interferometer with a
nonlinear phase shifter $S\left(\phi,k\right).$ The two input ports of this
interferometer are a coherent state $\left|\alpha\right\rangle_{a}$ and a
vacuum state $\left|0\right\rangle_{b},$ respectively. OPA is an optical
parametric amplifier and Hom is a homodyne detection.

##  II Nonlinear phase estimation model

Let us begin with the description of nonlinear phase estimation model, as
shown in Fig. 1(b). The nonlinear interferometer consists of two OPAs (or
FWMs) and a Kerr-type medium. Here we consider a CS
$\left|\alpha\right\rangle_{a}$ with
$\alpha=\left|\alpha\right|e^{i\theta_{\alpha}}$ and a VS
$\left|0\right\rangle_{b}$ as the inputs in mode $a$ and mode $b$,
respectively. Thus the probing state can be shown as
$\left|\psi_{in}\right\rangle=$
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}.$ After going
through the first OPA, the resulting state is given by
$\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ i.e., a two-mode
squeezed CS, where the operator
$\hat{S}\left(\xi_{1}\right)=\exp(\xi_{1}^{\ast}\hat{a}\hat{b}-\xi_{1}\hat{a}^{\dagger}\hat{b}^{\dagger})$
represents the OPA process, $\xi_{1}=g_{1}e^{i\theta_{1}}$, with a gain factor
$g_{1}$ and a phase shift $\theta_{1}$ and
$\hat{a}(\hat{a}^{\dagger}),\hat{b}(\hat{b}^{\dagger})$ are the annihilation
(creation) operators, for modes $a$ and $b$, respectively. For simplicity, we
assume that the Kerr-type medium is inset into the path $b$ between the first
and second OPAs to generate a nonlinear phase shift $\phi$ to be estimated.

After the interaction between the state
$\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$ with the Kerr-type
medium, the corresponding modified state becomes

$\left|\psi_{\phi}\right\rangle=\hat{S}\left(\phi,k\right)\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle,$
(1)

depending on the phase parameter $\phi$, where

$\hat{S}\left(\phi,k\right)=e^{i\phi(\hat{b}^{\dagger}\hat{b})^{k}},$ (2)

is the nonlinear phase shift operator and the exponent $k$ is the order of the
nonlinearity. In particular, for the case of $k=1$,
$\hat{S}\left(\phi,1\right)=e^{i\phi\hat{b}^{\dagger}\hat{b}}$ just reduces to
the linear phase shift, while for the case of $k=2$,
$\hat{S}\left(\phi,2\right)=e^{i\phi(\hat{b}^{\dagger}\hat{b})^{2}}$
corresponds to Kerr nonlinear case. Throughout this paper, we only consider
both Kerr-type nonlinear medium and linear one by taking $k=1,2$,
respectively.

After the second OPA, the final output state is given by

$\left|\psi_{out}\right\rangle=\hat{S}\left(\xi_{2}\right)\hat{S}\left(\phi,k\right)\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle,$
(3)

where
$\hat{S}\left(\xi_{2}\right)=\exp(\xi_{2}^{\ast}\hat{a}\hat{b}-\xi_{2}\hat{a}^{\dagger}\hat{b}^{\dagger}),$
with $\xi_{2}=g_{2}e^{i\theta_{2}},$ is a two-mode squeezing operator,
corresponding to the second OPA process. The Kerr nonlinear phase shifter
$\hat{S}\left(\phi,2\right)$ satisfying the transformation relation:

$\hat{S}^{\dagger}\left(\phi,2\right)\hat{b}^{\dagger}\hat{S}\left(\phi,2\right)=e^{-i\phi}\hat{b}^{\dagger}e^{-i2\phi\hat{b}^{\dagger}\hat{b}},$
(4)

which is useful for the calculation of phase sensitivity. One can refer to
Appendix A about more details of derivation for this relation. Finally, the
homdyne detection is performed on mode $a$, so that one can read information
about the value of $\phi$.

Figure 2: (Color online) (a) Phase sensitivity based on homodyne detection and
(b) the output signal as a function of $\phi$ with $g=1$,
$\left|\alpha\right|=1$, and $\theta_{\alpha}=\frac{\pi}{2}$. The black-dashed
line and the blue-solid line correspond to the linear phase shift ($k=1$) and
the Kerr nonlinear phase shift ($k=2$), respectively.

Figure 3: (Color online) Phase sensitivity based on homodyne detection as a
function of the gain factor $g$ and the coherent amplitude
$\left|\alpha\right|$ with $\theta_{\alpha}=\pi/2$ and $\phi=0$ for (a) the
linear phase shift ($k=1$), and (b) the Kerr nonlinear phase shift ($k=2$),
respectively.

##  III Phase sensitivity via homodyne detection

Next, we investigate the phase sensitivity of nonlinear phase. For this
purpose, we need to choose a special detection method for the readout of phase
information at the final output port. Actually, there are many different
detection methods, such as homodyne detection 11 ; 31 ; 32 , intensity
detection 33 ; 34 ; 47 , and parity detection 27 ; 35 ; 36 . Each way of
measurement has its own advantages. For example, the parity detection has been
proved to be an optimal detection for linear phase estimation in lots of
schemes 27 ; 51 . Compared with both intensity and parity detections, however,
homodyne detection can be easily realized with current technologies, thereby
playing a key role in the field of continuous-variable quantum key
distribution (CV-QKD) 52 ; 53 ; 54 ; 55 ; 56 . Therefore, in our scheme the
homodyne detection is employed on mode $a$ at one of output port to estimate
the phase parameter $\phi,$ where the detected variable is the amplitude
quadrature $\hat{X}$, i.e.,

$\hat{X}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}.$ (5)

Based on the error propagation formula, the phase sensitivity can be
calculated by

$\Delta\phi=\frac{\sqrt{\Delta^{2}\hat{X}}}{\left|\partial\left\langle\hat{X}\right\rangle/\partial\phi\right|},$
(6)

with
$\Delta^{2}\hat{X}=\left\langle\hat{X}^{2}\right\rangle-\left\langle\hat{X}\right\rangle^{2}.$
According to Eq. (6), for an arbitrary value of $\phi$, the corresponding
phase sensitivity can be analytically derived. For simplicity, the
corresponding expression is not shown here. One can refer to Appendix B for
more details. In the following discussions, we assume that the KSU(1,1)
interferometer is in a balanced situation, i.e., $\theta_{2}-\theta_{1}=\pi$
and $g_{1}=g_{2}=g.$

In Fig. 2(a), we show the phase sensitivity changing with $\phi$ for the
linear ($k=1$) and nonlinear ($k=2$) phase shifts when given parameters
$g=1,|\alpha|=1,$and $\theta_{\alpha}=\pi/2.$ It is clearly seen that, for
both cases above, the minimum value of the phase sensitivity can be achieved
at the optimal point $\phi=0.$ In addition, the phase sensitivity for $k=2$ is
always significantly superior to that for $k=1$ around the optimal point. This
implies that the nonlinear phase shift can be further used to improve the
phase sensitivity, compared to the linear one. The reason lies in that the
introduction of nonlinear phase shift increases the slope
$\partial\left\langle\hat{X}\right\rangle/\partial\phi$ of the output signal
$\left\langle\hat{X}\right\rangle$ at $\phi=0,$ which leads to the increase of
the denominator in Eq. (6), as shown in Fig. 2(b). In fact, this point will be
clear by deriving the phase sensitivity $\Delta\phi_{k}$ for $k=1,2$ at
$\phi=0$, which are given by

$\displaystyle\Delta\phi_{1}$
$\displaystyle=\frac{1}{\sqrt{N_{\alpha}}N_{OPA}},$ (7)
$\displaystyle\Delta\phi_{2}$
$\displaystyle=\frac{\Delta\phi_{1}}{1+N_{OPA}\left(N_{\alpha}+2\right)},$ (8)

where $\Delta\phi_{1}$ represents the phase sensitivity of the traditional
SU(1,1) interferometer with a linear phase shift $k=1,$
$N_{\alpha}=\left|\alpha\right|^{2}$ is the mean photon number of the input
coherent state, and $N_{OPA}=2\sinh^{2}g$ is the total photon number after the
first OPA with vacuum inputs. Obviously, $\Delta\phi_{2}>\Delta\phi_{1}.$

Moreover, from Fig. 2(b) it is shown that the peak width for the case of $k=2$
is narrower than that for the case of $k=1.$ In this sense, the nonlinear
phase shift can also improve supperresolution compared to the linear one. On
the other hand, to display the effects of both the coherence amplitude
$\left|\alpha\right|$ and the gain factor $g$ on the phase sensitivity, at the
optimal point $\phi=0,$ we also plot the phase sensitivity as a function of
$\left|\alpha\right|$ and $g$ with different values of $k=1,2$, in Figs. 3(a)
and 3(b), respectively. It is found that, the values of $\Delta\phi$ rapidly
decrease with the increase of $\left|\alpha\right|$ and $g$, especially for
the case of $k=2.$

Figure 4: (Color online) Phase sensitivity based on homodyne detection as a
function of (a) the gain factor $g$ with $\left|\alpha\right|=1$. (b) the
coherent amplitude $\left|\alpha\right|$ with $g=1$ ($\theta_{\alpha}=\pi/2$
). The black-dashed and blue-solid lines are respectively the linear phase
shift ($k=1$), and the Kerr nonlinear phase shift ($k=2$), while the purple-,
green- and red-solid lines respectively correspond to the SQL, HL and SHL.
Figure 5: (Color online) Phase sensitivity based on homodyne detection as a
function of the total average photon number N of input state with $g=1.$ The
orange- and cyan-dot-dashed lines respectively correspond to a squeezed vacuum
state combined with a coherent states input and two coherent states with the
linear phase shift ($k=1$), as shown in Ref. 31 ; 32 ; while the black-dashed
and blue-solid lines are respectively the linear phase shift ($k=1$), and the
Kerr nonlinear phase shift ($k=2$).

To further show the advantage of our scheme, we also make a comparison about
phase sensitivities, involving the SQL ($\Delta\phi\thicksim
1/\sqrt{N_{Total}}$), the HL ($\Delta\phi\thicksim 1/N_{Total}$) and the Super
HL (SHL) ($\Delta\phi\thicksim 1/N_{Total}^{2}$), as shown in Fig. 4. Note
that
$N_{Total}=\left\langle\psi_{in}\right|\hat{S}^{\dagger}\left(\xi_{1}\right)(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b})\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$
is the total mean photon number inside the KSU(1,1) interferometer. From Fig.
4, the black dashed line is the sensitivity performance of the linear phase
shift, which can only break through SQL and is always surpassed by that of the
Kerr nonlinear phase shift. In particular, the latter can break both the SQL
and the HL, but cannot beat the SHL. The reason may be that adopting the Kerr
nonlinear phase shift to effectively improve the maximum amount of information
about the unknown phase shift $\phi$ can result in a transition from HL
($\Delta\phi\thicksim 1/N_{Total}$) to SHL ($\Delta\phi\thicksim
1/N_{Total}^{2}$), as described in Refs. 9 ; 57 ; 58 ; 59 ; 60 ; 61 ; 62 .
Nevertheless, the common feature of these two is that the phase sensitivity
increases with the increase of $\left|\alpha\right|$ and $g$.

Next, we make a comparison about the performance of phase sensitivity. In Fig.
5, the phase sensitivity is plotted as a function of the total average input
photon number $N$ with several different input resources, including
$\left|\alpha\right\rangle_{a}\otimes\left|\beta\right\rangle_{b}$,
$\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$ and
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ used in the
traditional SU(1,1) interferometer ($k=1$) and
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ used in the
KSU(1,1) interferometer ($k=2$). It is shown that, for the traditional case,
the phase sensitivity with the input
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ performs the
worst among these three inputs 31 ; 32 . This implies that the introduction of
coherence amplitude $\left|\alpha\right|$ or squeezing parameter $\varsigma$
is beneficial for the phase sensitivity improvement. However, for the KSU(1,1)
interferometer, it is interesting that the best phase sensitivity can be
achieved only using the simplest input
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$, which is
significantly superior to those inputs in the traditional SU(1,1)
interferometer. That is to say, for a simple input with less energy and less
resources, the phase sensitivity can be further improved by introducing Kerr
nonlinear phase.

Figure 6: (Color online) (a) The quantum Fisher information $F$ and (b) the
QCRB as a function of gain factor g for $\left|\alpha\right|=1$ (
$\theta_{\alpha}=\pi/2$), respectively. The black-dashed and blue-solid lines
correspond to linear phase shift ($k=1$) and Kerr nonlinear phase shift
($k=2$), respectively. Figure 7: (Color online) The $\Delta\phi_{QCRB}$ as a
function of the total average photon number N of input state with $g=1.$ The
orange and cyan lines correspond to a coherent state plus a squeezed vacuum
state input and two coherent states, respectively, with the linear phase shift
($k=1$); while the black and blue lines correspond to $k=1$, and the Kerr
nonlinear phase shift ($k=2$), respectively. The dashed and solid lines
correspond to the $\Delta\phi_{QCRB}$ and the $\Delta\phi,$ respectively.

##  IV The QFI in the KSU(1,1) interferometer

As an elegant approach, the QFI can be used to visually describe the maximum
amount of information about the unknown phase shift $\phi$, which is connected
with the QCRB. In fact, the QFI is the intrinsic information in quantum state
and is independent of any specific detection scheme. In the absence of losses,
for a pure state, the corresponding QFI can be calculated as 34 ; 48 ; 49

$F=4\left[\left\langle\psi_{\phi}^{\prime}|\psi_{\phi}^{\prime}\right\rangle-\left|\left\langle\psi_{\phi}^{\prime}|\psi_{\phi}\right\rangle\right|^{2}\right],$
(9)

where
$\left|\psi_{\phi}\right\rangle=\hat{S}\left(\phi,k\right)\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$
is the state vector prior to the second OPA and
$\left|\psi_{\phi}^{\prime}\right\rangle=\partial\left|\psi_{\phi}\right\rangle/\partial\phi.$
Then, for the linear phase shift $\left(k=1\right)$ and the Kerr nonlinear
phase shift $\left(k=2\right),$ the QFI can be, respectively, reformed as 45

$F_{1}=4\left\langle\Delta^{2}\hat{n}\right\rangle,F_{2}=4\left\langle\Delta^{2}\hat{n}^{2}\right\rangle,$
(10)

where $\hat{n}=\hat{b}^{\dagger}\hat{b}$ is the photon number operator on mode
$b$ and $\left\langle\Delta^{2}\hat{n}^{j}\right\rangle=$
$\left\langle\bar{\psi}_{out}\right|\left(\hat{n}^{j}\right)^{2}\left|\bar{\psi}_{out}\right\rangle-\left\langle\bar{\psi}_{out}\right|\hat{n}^{j}\left|\bar{\psi}_{out}\right\rangle^{2},\left(j=1,2\right)$
with the state vector after the first OPA, i.e.,
$\left|\bar{\psi}_{out}\right\rangle=\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle.$
Using the normal ordering forms of the operators,

$\displaystyle(\hat{b}^{\dagger}\hat{b})^{4}$ $\displaystyle=\hat{b}^{\dagger
4}\hat{b}^{4}+6\hat{b}^{\dagger 3}\hat{b}^{3}+7\hat{b}^{\dagger
2}\hat{b}^{2}+\hat{b}^{\dagger}\hat{b},$
$\displaystyle(\hat{b}^{\dagger}\hat{b})^{2}$ $\displaystyle=\hat{b}^{\dagger
2}\hat{b}^{2}+\hat{b}^{\dagger}\hat{b},$ (11)

and the characteristic function method for calculating average, it is ready to
have

$F_{1}=4\left(\bar{A}_{2}+\bar{A}_{1}-\bar{A}_{1}^{2}\right),F_{2}=F_{1}+f,$
(12)

with

$\displaystyle f$
$\displaystyle=4\left[\bar{A}_{4}+6\left(\bar{A}_{3}+\bar{A}_{2}\right)-\bar{A}_{2}\left(\bar{A}_{2}+2\bar{A}_{1}\right)\right],$
(13) $\displaystyle\bar{A}_{m}$
$\displaystyle=m!(\sinh^{2m}g)L_{m}(-\left|\alpha\right|^{2}),m\in\\{1,2,3,4\\},$
(14)

and $L_{m}(\bullet)$ is the Laguerre polynomials. One can refer to Appendix C
for more details about derivations of $\bar{A}_{m}$. Based on Eq. (12), we can
give the QCRB $\Delta\phi_{QCRB},$ which represents the lower bound of the
phase sensitivity, i.e.,

$\Delta\phi_{QCRB}=\frac{1}{\sqrt{\upsilon F_{k}}},\text{
}\left(k=1,2\right),$ (15)

where $\upsilon$ is the number of trials. For simplicity, here we set
$\upsilon=1.$ In general, the smaller the values of $\Delta\phi_{QCRB}$, the
higher the phase sensitivity.

To clearly see this point, according to Eqs. (12) and (15), at a fixed
$\left|\alpha\right|=1,$ Fig. 6 shows the QFI and the QCRB changing with $g$
for different $k=1,2.$ It is clear that, for the cases of $k=1,2,$ the values
of the QFI increase significantly with the increase of $g,$ thereby leading to
the more precise phase sensitivity. Moreover, when given the same gain factor
$g$, both the QFI and the QCRB of $k=2$ always outperform those of $k=1$,
which distinctly shows the superiority of the nonlinear phase shift compared
to the linear case. This result originates from the additional item $f$ in Eq.
(12), giving rise to the increase of QFI.

As a comparison, we also consider the $\Delta\phi_{QCRB}$ as a function of $N$
for several different input resources in Fig. 7, similar to the phase
sensitivity (in Fig. 5). Some similar results are obtained. For instance,
although the QCRB with
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ is the worst
than that with other resources in the traditional SU(1,1) interferometer 63 ,
the smallest $\Delta\phi_{QCRB}$ can be realized by the simplest input
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the KSU(1,1)
interferometer. In addition, comparing Fig. 5 with Fig. 7, it is found that
compared with several different input resources in the traditional SU(1,1)
interferometer, the input
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the KSU(1,1)
interferometer is closer to the ultimate phase precision $\Delta\phi_{QCRB}.$

##  V The effects of photon losses

In the realistic scenario, the decoherence process is unavoidable because
there always exist interaction between the interferometer system and its
surrounding environment, thereby leading to the information leakage from the
system to the environment. For this season, the system performance would drop
severely. In general, there are many types of decoherence process 64 ; 65 ,
such as photon loss, imperfect detector, phase diffusion, and so on. Here, for
simplicity, we only study the effects of photon losses on both the phase
sensitivity and the QFI in our scheme.

Figure 8: (Color online) Schematic diagram of the photon losses that occur
before and after the second OPA. $T_{j}\left(j=1,2\right)$ is the
transmissivity of the fictitious beam splitters; $a_{v_{j}}$ and $b_{v_{j}}$
are the vacuum noise operators.

###  V.1 The effects of photon losses on the phase sensitivity

Figure 9: (Color online) Phase sensitivity based on homodyne detection as a
function of the photon losses inside the interferometer $T_{1}$ and the photon
losses outside the interferometer $T_{2}$ with $g=1$ and
$\left|\alpha\right|=1$ ($\theta_{\alpha}=\pi/2$). (a) and (b) correspond to
the cases with$k=1,2$, respectively.
Figure 10: (Color online) The output signal as a function of $\phi$ with
$g=1$, $\left|\alpha\right|=1$, $\theta_{\alpha}=\frac{\pi}{2},$and
$T_{1}=T_{2}=0.6$. The black and blue lines correspond to $k=1,2$,
respectively. The dashed and solid lines correspond to the effects of photon
losses and no photon losses, respectively.

First, let us consider the effects of photon losses before and after the
second OPA on the phase sensitivity. Actually, the losses after the second OPA
can also be seen as the detection imperfect at out ports. To describe the
photon losses, the channel is usually simulated by inserting the fictitious
beam splitter (BSj) with transmissivity $T_{j}\left(j=1,2\right),$ as shown in
Fig. 8. Generally speaking, the smaller the values of $T_{j}$, the more
serious the photon loss. Here, we denote the photon loss inside and outside
the interferometer as internal loss ($T_{1}$) and external one ($T_{2}$),
respectively. In the following discussion, we assume that the losses of
different modes are the same inside or outside the interferometer.

For any single-mode pure state $\left|\varphi\right\rangle_{in}$, after
through the photon loss, the output state $\left|\varphi\right\rangle_{out}$
in an enlarged space can be expressed as a pure state, i.e.,
$\left|\varphi\right\rangle_{out}=\hat{U}_{BS_{j}}\left|\varphi\right\rangle_{in}\left|0\right\rangle$.
Thus, the average value of operator $\hat{O}$ can be calculated as
$\left\langle\hat{O}\right\rangle=\left.\left\langle
0\right|_{in}\left\langle\varphi\right|\right.\hat{U}_{BS_{j}}^{\dagger}\hat{O}\hat{U}_{BS_{j}}\left.\left|\varphi\right\rangle_{in}\left|0\right\rangle\right.$,
which is equivalent to the average value in the reduced density operator
$\rho_{out}=$Tr$[\hat{U}_{BS_{j}}\left|\varphi\right\rangle_{in}\left|0\right\rangle\left\langle
0\right|_{in}\left\langle\varphi\right|\hat{U}_{BS_{j}}^{\dagger}]$. In our
scheme in Fig. 8, the final output state $\left|\varphi\right\rangle_{f}$ in
the enlarged space is given by

$\left|\varphi\right\rangle_{f}=\hat{U}_{BS_{2}}^{b}\hat{U}_{BS_{2}}^{a}\hat{S}\left(\xi_{2}\right)\hat{U}_{BS_{1}}^{b}\hat{U}_{BS_{1}}^{a}\left|\psi_{\phi}\right\rangle\left|\vec{0}\right\rangle,$
(16)

where $\left|\vec{0}\right\rangle=\left|0000\right\rangle$ is the noise vacua,
and
$\hat{U}_{BS_{j}}^{a}=\exp[\arccos\sqrt{T_{j}}(\hat{a}^{\dagger}\hat{a}_{\upsilon_{j}}-\hat{a}\hat{a}_{\upsilon_{j}}^{\dagger})]$
and
$\hat{U}_{BS_{l}}^{b}=\exp[\arccos\sqrt{T_{l}}(\hat{b}^{\dagger}\hat{b}_{\upsilon_{l}}-\hat{b}\hat{b}_{\upsilon_{l}}^{\dagger})]$
are the BS operators acting on mode $\hat{a}$ and $\hat{b}$, respectively.
Here $\hat{a}_{\upsilon_{j}}$ and $\hat{b}_{\upsilon_{l}}$ are the vacuum
noise operators.

Using the transformations of the fictitious BSj on the annihilation operators,
i.e.,

$\displaystyle(\hat{U}_{BS_{j}}^{a})^{\dagger}\hat{a}\hat{U}_{BS_{j}}^{a}$
$\displaystyle=\sqrt{T_{j}}\hat{a}+\sqrt{1-T_{j}}\hat{a}_{\upsilon_{j}},$ (17)
$\displaystyle(\hat{U}_{BS_{l}}^{b})^{\dagger}\hat{b}\hat{U}_{BS_{l}}$
$\displaystyle=\sqrt{T_{l}}\hat{b}+\sqrt{1-T_{l}}\hat{b}_{\upsilon_{l}},$ (18)

it is not difficult to obtain the phase sensitivity $\Delta\phi_{L_{k}}$
($k=1,2$) after losses, not shown here for simplicity. In particular, at the
optimal point $\phi=0,$ for the traditional and KSU(1,1) interferometers in
the presence of photon loss, the phase sensitivities $\Delta\phi_{L_{k}}$
$(k=1,2)$ are derived, respectively, as

$\Delta\phi_{L_{1}}=\sqrt{\left(\Delta\phi_{1}\right)^{2}+\frac{\left(1-T_{1}\right)T_{2}\cosh
2g+1-T_{2}}{4T_{1}T_{2}\left|\alpha\right|^{2}\sinh^{4}g}},$ (19)

and

$\Delta\phi_{L_{2}}=\frac{\Delta\phi_{L_{1}}}{1+N_{OPA}\left(N_{\alpha}+2\right)},$
(20)

where the phase sensitivity $\Delta\phi_{L_{1}}$ corresponds to that of the
traditional SU(1,1) interferometer in the presence of photon loss. In Eq.
(19), the first term $\left(\Delta\phi_{1}\right)^{2}$ can be given by Eq.
(8), while the second term is resulted from the internal and external losses.
From Eqs. (19) and (20), it is obvious that
$\Delta\phi_{L_{2}}>\Delta\phi_{L_{1}}\ $in the presence of losses.

Figure 11: (Color online) Phase sensitivity based on homodyne detection as a
function of (a) the gain factor $g$ with $\left|\alpha\right|=1$, (b) the
coherent amplitude $\left|\alpha\right|$ with $g=1$ $(\theta_{\alpha}=\pi/2).$
The black and blue lines correspond to $k=1,2$, respectively. The dot-dashed
line, dashed line, and solid line correspond to the photon losses inside the
interferometer ($T_{1}=0.6$ and $T_{2}=1$), outside the interferometer
($T_{1}=1$ and $T_{2}=0.6$), and no photon losses ($T_{1}=$ $T_{2}=1$),
respectively.

In order to visually see the influence of losses, at fixed values of $g=1$ and
$\left|\alpha\right|=1,$ we plot the phase sensitivity $\Delta\phi_{L_{k}}$
$\left(k=1,2\right)$ as a function of $T_{j}\left(j=1,2\right)$ for different
values of $k=1,2,$ as shown in Fig. 9. It is seen that for $k=1,2,$ the values
of $\Delta\phi_{L_{k}}$ increase with the decrease of $T_{j},$ which shows
that the phase sensitivity can be deeply influenced by the photon losses.
Compared to the case of $k=1$ (Fig. 9(a)), the phase sensitivity of $k=2$
(Fig. 9(b)) is more robust against photon losses under the same conditions.
The reason lies in that the Kerr nonlinear phase shift $\left(k=2\right)$ is
beneficial to the increase of the denominator in the error propagation formula
(see Fig. 10), thereby resulting in the improvement of the phase sensitivity
even in the presence of the photon losses. In addition, from Fig. 10, it is
also found that the full width of the main peaks for $k=2$ would broaden at
fixed photon-loss factors $T_{j}=0.6$, which implies that the signal’s super-
resolution characteristic is going to be worse. Even so, the usage of the Kerr
nonlinear phase shift performs better than that of the linear one with respect
to increasing the signal’s super-resolution.

Figure 12: (Color online) Phase sensitivity based on homodyne detection as a
function of the total average photon number N of input state with $g=1,$
$T_{1}=T_{2}=0.6.$ The orange- and cyan-dot-dashed lines respectively
correspond to a squeezed vacuum state plus a coherent states input and two
coherent states with $k=1$, as shown in Ref. 31 ; 32 ; The black and blue
lines correspond to $k=1,2$, respectively. The dashed and solid lines
correspond to the effects of photon losses and no photon losses, respectively.

Next, we consider their own effects of internal and external losses. From Eqs.
(20) and (19), for the internal and external losses, the phase sensitivity
$\Delta\phi_{L_{1}}$ and $\Delta\phi_{L_{2}}$ can be reduced, respectively, to

$\displaystyle\left.\Delta\phi_{L_{1}}\right|_{T_{2}=1}$
$\displaystyle=\sqrt{\left(\Delta\phi_{1}\right)^{2}+\frac{\left(1-T_{1}\right)\cosh
2g}{4T_{1}\left|\alpha\right|^{2}\sinh^{4}g}},$ (21)
$\displaystyle\left.\Delta\phi_{L_{2}}\right|_{T_{2}=1}$
$\displaystyle=\frac{\left.\Delta\phi_{L_{1}}\right|_{T_{2}=1}}{1+N_{OPA}\left(N_{\alpha}+2\right)},$
(22)

and

$\displaystyle\left.\Delta\phi_{L_{1}}\right|_{T_{1}=1}$
$\displaystyle=\sqrt{\left(\Delta\phi_{1}\right)^{2}+\frac{1-T_{2}}{4T_{2}\left|\alpha\right|^{2}\sinh^{4}g}},$
(23) $\displaystyle\left.\Delta\phi_{L_{2}}\right|_{T_{1}=1}$
$\displaystyle=\frac{\left.\Delta\phi_{L_{1}}\right|_{T_{1}=1}}{1+N_{OPA}\left(N_{\alpha}+2\right)}.$
(24)

From Eqs. (21)-(24), it is clear that the effects of the internal losses are
greater than those of the external ones on the phase sensitivity due to
$\cosh\left(2g\right)>1$ for the same losses. To clearly see this point, Fig.
11 shows the phase sensitivity $\Delta\phi$ as a function of
$\left|\alpha\right|$ or $g$ for the internal and external losses. It is shown
that, for a fixed $k$, the gap between the ideal scenario and the internal-
losses one is larger than that between the ideal scenario and the external-
losses one. The gap increases first and then decreases, while the phase
sensitivity still increases as the $\left|\alpha\right|$ or $g$ increases. It
is interesting that, the gap for $k=2$ becomes significantly smaller than that
for $k=1$, which implies that the Kerr nonlinear phase shift can effectively
resist the influences of photon losses.

Before the end of this part, for some different input states, we make a
comparison about the phase sensitivity in the presence of losses. In Fig. 12,
we plot the $\Delta\phi$ as a function of total average input photon number
$N$ for those resources above, at fixed values $g=1$ and $T_{1}=T_{2}=0.6.$ It
is shown that, for the case of $k=1,$ the gap with
$\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$
(orange lines) between the ideal and loss cases is the largest, which means
that the CS plus the SVS as inputs are more sensitive to the photon losses
than other input resources 31 . To some extent, the problem sensitive to noise
can be solved by applying two CSs into the input ports of SU(1,1)
interferometer, see cyan lines in Fig. 12 32 . The worst performance is still
kept by CS+VS input in lossy channel as in ideal case. However, for the case
of $k=2$, the CS+VS input presents the best performance of phase sensitivity
(blue lines). Even in the presence of photon losses, the phase sensitivity
with the CS+VS is also significantly superior to that with other resources in
ideal case for $k=1$. This indicates that the Kerr nonlinear phase shift can
dramatically suppress the decoherence.

###  V.2 The effects of photon losses on the QFI

For a realistic case, the output state after lossy channel is usually a mixed
state not a pure one. Thus the QFI can not be directly discussed according to
Eq. (9). In this case, one may appeal to the spectral decompositions of
density operator 66 ; 67 . Generally speaking, this method is not only
difficult to obtain the spectral decompositions, but also to derive the
analytical expression of the QFI. Fortunately, Escher et al. proposed a way to
obtain an upper bound to the QFI in the presence of photon losses 68 . The
basic idea is to purify the whole system involving an initial pure state and
an environment by introducing additional degrees of freedom, so that the
present problem is converted to the parameter estimation under a unitary
evolution $\hat{U}_{S+E}\left(\phi\right)$. We shall use this way to obtain
the analytical QFI in a realistic case. For this purpose, we first make a
brief review in the following.

Given an initial pure state $\left|\psi_{S}\right\rangle$ in the probe system
$S$ and an initial state $\left|0_{E}\right\rangle$ in the environment, the
purified state $\left|\psi_{S+E}\right\rangle$ in the enlarged space can be
given by

$\displaystyle\left|\psi_{S+E}\right\rangle$
$\displaystyle=\hat{U}_{S+E}\left(\phi\right)\left|\psi_{S}\right\rangle\left|0_{E}\right\rangle,$
$\displaystyle=\sum_{l=0}^{\infty}\hat{\Pi}_{l}\left(\phi\right)\left|\psi_{S}\right\rangle\left|l_{E}\right\rangle$
(25)

where $\hat{\Pi}_{l}\left(\phi\right)$ is the Kraus operator describing the
photon losses (also including the effect of phase shift), and
$\left|l_{E}\right\rangle$ is an orthogonal basis of the state
$\left|0_{E}\right\rangle.$ In this situation, for the whole purified system,
the QFI
$C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
turns out to be

$\displaystyle
C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
$\displaystyle=4\left[\left\langle\psi_{S+E}^{\prime}|\psi_{S+E}^{\prime}\right\rangle-\left|\left\langle\psi_{S+E}^{\prime}|\psi_{S+E}\right\rangle\right|^{2}\right].$
(26)

According to Eqs. (25) and (26), the upper bound
$C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
can be given in terms of Kraus operators

$C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]=4\left[\left\langle\hat{H}_{1}\left(\phi\right)\right\rangle_{S}-\left\langle\hat{H}_{2}\left(\phi\right)\right\rangle_{S}^{2}\right],$
(27)

with the averages $\left\langle\cdot\right\rangle_{S}$ being derived in
$\left|\psi_{S}\right\rangle$ and

$\displaystyle\hat{H}_{1}$
$\displaystyle=\sum_{l=0}^{\infty}\frac{d\hat{\Pi}_{l}^{\dagger}\left(\phi\right)}{d\phi}\frac{d\hat{\Pi}_{l}\left(\phi\right)}{d\phi},$
(28) $\displaystyle\hat{H}_{2}$
$\displaystyle=i\sum_{l=0}^{\infty}\frac{d\hat{\Pi}_{l}^{\dagger}\left(\phi\right)}{d\phi}\hat{\Pi}_{l}\left(\phi\right).$
(29)

Actually, Eq. (27) provides an upper bound to the QFI $F_{L}\leqslant
C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
for the reduced system 62 , thus one need to find the minimal value over all
Kraus operators $\left\\{\Pi_{l}\left(\phi\right)\right\\}$, i.e.,

$F_{L}=\underset{\left\\{\Pi_{l}\left(\phi\right)\right\\}}{\min}C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right].$
(30)

Figure 13: (Color online) Schematic diagram of the photon losses that occur
before and after the nonlinear phase. $\eta$ is the transmissivity of the
fictitious BS; $b_{v}$ is the vacuum operator.

In our scheme, we consider the QFI of the KSU(1,1) interferometer with the
photon losses placed before or after the Kerr nonlinear phase shift in $b$
arm, as shown in Fig. 13. The corresponding Kraus operator
$\hat{\Pi}_{l}\left(\phi\right)$ including the nonlinear phase can be written
as the general form

$\hat{\Pi}_{l}\left(\phi\right)=\sqrt{\frac{\left(1-\eta\right)^{l}}{l!}}e^{i\phi[(\hat{b}^{\dagger}\hat{b})^{2}-2\mu_{1}\hat{b}^{\dagger}\hat{b}l-\mu_{2}l^{2}]}\eta^{\frac{\hat{n}}{2}}\hat{b}^{l},$
(31)

where $\eta$ denotes the strength of the photon loss. $\eta=0$ and $\eta=1$
describe the complete absorption and lossless cases, respectively. $\mu_{1}$
and $\mu_{2}$ are two variational parameters with $\mu_{1}=\mu_{2}=0$ and
$\mu_{1}=\mu_{2}=-1$ corresponding to the photon losses occurring before and
after the Kerr nonlinear phase shift, respectively.

To derive Eq. (27) using Eq. (31), we shall appeal to the technique of
integration within an ordered product of operators (IWOP) 69  to derive the
operator identity [see Appendix D], i.e.,

$\eta^{\hat{n}}\hat{n}^{q}=\colon\left.\frac{\partial^{q}}{\partial
x^{q}}e^{\left(\eta
e^{x}-1\right)\hat{b}^{\dagger}\hat{b}}\right|_{x=0}\colon,$ (32)

where $:\cdot:$ indicates the symbol of the normal ordering form, which
further leads to the formula [see Appendix D]

$\displaystyle\sum_{l=0}^{\infty}\frac{\left(1-\eta\right)^{l}}{l!}l^{p}\hat{b}^{\dagger
l}\eta^{\hat{n}}\hat{n}^{q}\hat{b}^{l}$ $\displaystyle=D_{q,p}\left[\eta
e^{x}+\left(1-\eta\right)e^{y}\right]^{\hat{b}^{\dagger}\hat{b}},$ (33)

with $D_{q,p}=\frac{\partial^{q+p}}{\partial x^{q}\partial
y^{p}}\left[\cdot\right]_{|x=y=0}$ being a partial differential operator.

Using Eq. (33) and the following transformation relations

$\displaystyle
e^{\lambda\hat{b}^{\dagger}\hat{b}}\hat{b}^{l}e^{-\lambda\hat{b}^{\dagger}\hat{b}}$
$\displaystyle=e^{-\lambda l}\hat{b}^{l},$ $\displaystyle
e^{\lambda(\hat{b}^{\dagger}\hat{b})^{2}}\hat{b}^{l}e^{-\lambda(\hat{b}^{\dagger}\hat{b})^{2}}$
$\displaystyle=e^{\lambda l^{2}}\hat{b}^{l}e^{-2\lambda
l\hat{b}^{\dagger}\hat{b}},$ (34)

the upper bound of the QFI
$C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
can be calculated as [see Appendix E]

$\displaystyle
C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
$\displaystyle=4(W_{1}^{2}\left\langle\Delta^{2}\hat{n}^{2}\right\rangle-
W_{2}\left\langle\hat{n}^{3}\right\rangle+W_{3}\left\langle\hat{n}^{2}\right\rangle$
$\displaystyle-W_{4}\left\langle\hat{n}\right\rangle-
W_{5}\left\langle\hat{n}^{2}\right\rangle\left\langle\hat{n}\right\rangle-
W_{6}\left\langle\hat{n}\right\rangle^{2}),$ (35)

where $\left\langle\cdot\right\rangle$ and
$\left\langle\Delta^{2}\cdot\right\rangle$ are, respectively, the average and
variance under the state $\left|\psi_{S}\right\rangle,$ where
$\left|\psi_{S}\right\rangle=S\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$
is the state vector after the first OPA. $W_{j}\left(j=1,2,3,4,5,6\right)$ are
given in Appendix E, not shown here for simplicity. In particular, when
$C_{Q}\left[\left|\psi_{S}\right\rangle,\hat{\Pi}_{l}\left(\phi\right)\right]$
reaches the minimum value corresponding to the QFI $F_{L}$, the variational
parameters $\mu_{1}$ and $\mu_{2}$ are, respectively, given by

$\displaystyle\mu_{1opt}$ $\displaystyle=\frac{BE-CD}{AD-2\eta B^{2}},$ (36)
$\displaystyle\mu_{2opt}$ $\displaystyle=\frac{AE-2\eta BC}{AD-2\eta B^{2}},$
(37)

where $A,B,C,D$ and $E$ are shown in Appendix E. Upon substituting those
optimal results $\mu_{1opt}$ and $\mu_{2opt}$ into $C_{Q},$ the minimum value
of $C_{Q}$, i.e., the QFI $F_{L}$ of Kerr nonlinear phase shift in the
presence of photon losses, can be derived theoretically.

Figure 14: (Color online) The $\Delta\phi_{QCRB}$ as a function of (a) the
transmissivity $\eta$ with $g=1$ and $\left|\alpha\right|=1$, (b) the gain
factor $g$ with $\left|\alpha\right|=1.$ The black and blue lines correspond
to $k=1$,$2$, and the dashed and solid lines to the effects of photon losses
and no photon losses, respectively.

Considering a CS and a VS input, here we numerically analyze the QCRB
$\Delta\phi_{QCRB}$ which actually is equivalent to the QFI due to the
relation in Eq. (15). For fixed parameters with $|\alpha|=1$ and
$\theta_{\alpha}=\pi/2,$ the QCRB $\Delta\phi_{QCRB}$ as a function of
transmissivity $\eta$ and gain factor $g$ are shown in Fig. 14. For
comparison, the linear case with $k=1$ is also plotted here. It is clear that
the bound performance of phase sensitivity becomes better and better as the
increase of both transmissivity $\eta$ and gain factor $g$ for $k=1,2$ (see
dashed lines in Fig. 14 (a) and (b)). The QCRB for $k=2$ always outperforms
that for $k=1,$ and the gap between them first increase and then decrease as
the increase of $\eta$ and $g$. Compared to the ideal cases (solid lines in
Fig. 14(b)), it is clear that photon losses present an obvious effect on the
QCRB (dashed lines in Fig. 14(b)). However, it is interesting that the gap of
QCRB for $k=2$ between ideal and nonideal cases is significantly smaller than
that for $k=1$, which are also going to be smaller as the increase of $g$
especially for $k=2$. Again, this implies that the combination of Kerr
nonlinear and OPA can further compress the decoherence effect from the
environments.

Figure 15: (Color online) The $\Delta\phi_{QCRB}$ as a function of the total
average photon number N of input state with $g=1,$ $\eta=0.6.$ The orange- and
cyan-dot-dashed lines respectively correspond to a squeezed vacuum state plus
a coherent states input and two coherent states with $k=1$, as shown in Ref.
63 . The black and blue lines correspond respectively to $k=1$,2. The dashed
and solid lines correspond to the effects of photon losses and no photon
losses, respectively.

In addition, under the photon-loss processes (e.g., $\eta=0.6$), we further
consider the $\Delta\phi_{QCRB}$ as a function of total average input photon
number $N$ for those different input resources above, when given parameter
$g=1,$ as shown in Fig. 15. Here the solid lines correspond to the ideal cases
for a comparison. It is clearly seen that, compared to other input resources,
the $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ input
presents the worst QCRB in the linear phase shift, with and without photon
losses. However, thanks to the introduction of Kerr nonlinear phase, the QCRB
for $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ input is
the best, which is superior to, even in nonideal case, the QCRB in ideal case
with other inputs. This means that our scheme with and without photon losses
shows an obvious advantage of low-cost input resources to obtain a better QCRB
by introducing nonlinear phase instruments in the SU(1,1) interferometer
system.

##  VI Conclusions

In summary, we propose a protocol of second-order nonlinear phase estimation
by introducing the Kerr medium into the traditional SU(1,1) interferometer. A
kind of simplest inputs, coherent state plus vacuum state, and homodyne
detection for one mode are used. Both the phase sensitivity and the QFI or
QCRB are analytically derived in both ideal and nonideal scenario by using the
CF method and the IWOP technique. It is shown that the increase of both gain
factor $g$ and coherent amplitude $\left|\alpha\right|$ is beneficial for
improving both the phase sensitivity and the QCRB. Compared to the linear
phase estimation, our scheme presents a significantly better performance about
both of them, especially around the optimal point $\phi$ $=0$. In particular,
for the ideal case, our scheme breaks through both the SQL and the HL, even
approaching the SHL, at the large $g$ and $\left|\alpha\right|$ levels. In
addition, we consider the effects of the decoherence on the phase sensitivity,
from the internal or external photon losses. It is found that the former has a
more obvious decoherence than the latter. Even so, our scheme still provides a
significant improvement for the performance of phase estimation by homodyne
detection, compared to the linear case.

To further show the advantages of our scheme, with and without photon losses,
we also consider both the phase sensitivity and the QCRB changing with total
average input photon number $N$ for several different input resources,
including $\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b},$
$\left|\alpha\right\rangle_{a}\otimes\left|\beta\right\rangle_{b}$ and
$\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$, when
given parameters $g=1.$ It is found that the best phase sensitivity can be
achieved only using the simplest input
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ in the proposed
KSU(1,1) interferometer, which is significantly superior to those inputs in
the traditional SU(1,1) interferometer. This means that, for a simple input
with less energy and less resources, both the phase sensitivity and the QCRB
can be further improved by introducing Kerr nonlinear phase. These results may
have important applications in other quantum information processing.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 11964013, 11664017), the Training Program for Academic and
Technical Leaders of Major Disciplines in Jiangxi Province (20204BCJL22053),
the Postgraduate Scientific Research Innovation Project of Hunan Province
(Grant No. CX20190126) and the Postgraduate Independent Exploration and
Innovation Project of Central South University (Grant No. 2019zzts070).

Appendix A: Proof of the transformation relation

For a completeness, here, we give the proof about the transformation relation,
i.e.,
$\hat{S}^{\dagger}\left(\phi,2\right)\hat{b}^{\dagger}\hat{S}\left(\phi,2\right)=e^{-i\phi}\hat{b}^{\dagger}e^{-i2\phi\hat{b}^{\dagger}\hat{b}}$.
It is well known that any operator $\hat{\zeta}$ can be expanded in Fock state
space,

$\hat{\zeta}=\underset{m,n=0}{\overset{\infty}{\sum}}\zeta_{m,n}\left|m\right\rangle\left\langle
n\right|,$ (A1)

where $\zeta_{m,n}=\left\langle m\right|\hat{\zeta}\left|n\right\rangle$ is
the matrix element $\hat{\zeta}$ in Fock space.

Thus, if we take

$\displaystyle\hat{\zeta}$
$\displaystyle=\hat{S}^{\dagger}\left(\phi,2\right)\hat{b}^{\dagger}\hat{S}\left(\phi,2\right)$
$\displaystyle=e^{-i\phi\left(\hat{b}^{\dagger}\hat{b}\right)^{2}}\hat{b}^{\dagger}e^{i\phi\left(\hat{b}^{\dagger}\hat{b}\right)^{2}},$
(A2)

then the matrix element $\zeta_{m,n}$ can be calculated as

$\zeta_{m,n}=\sqrt{n+1}e^{-i\phi\left(2n+1\right)}\delta_{m,n+1},$ (A3)

where we have used the
$\hat{b}^{\dagger}\hat{b}\left|n\right\rangle=n\left|n\right\rangle$ and
$\hat{b}^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle.$
Substituting Eq. (A3) into Eq.(A1), we can get

$\displaystyle\hat{\zeta}$
$\displaystyle=\hat{S}^{\dagger}\left(\phi,2\right)\hat{b}^{\dagger}\hat{S}\left(\phi,2\right)$
$\displaystyle=\underset{m,n=0}{\overset{\infty}{\sum}}\sqrt{n+1}e^{-i\phi\left(2n+1\right)}\delta_{m,n+1}\left|m\right\rangle\left\langle
n\right|$
$\displaystyle=\underset{n=0}{\overset{\infty}{\sum}}\sqrt{n+1}e^{-i\phi\left(2n+1\right)}\left|n+1\right\rangle\left\langle
n\right|$
$\displaystyle=e^{-i\phi}\underset{n=0}{\overset{\infty}{\sum}}e^{-i2n\phi}\sqrt{n+1}\left|n+1\right\rangle\left\langle
n\right|$
$\displaystyle=e^{-i\phi}\hat{b}^{\dagger}e^{-i2\phi(\hat{b}^{\dagger}\hat{b})}.$
(A4)

Appendix B: Phase estimation based on homodyne measurement

Combining Eqs. (3) and (5), we can derive the standard deviation
$\Delta\hat{X}$ as

$\Delta\hat{X}=\sqrt{\left|\overline{U}\right|^{2}+\left|\overline{V}\right|^{2}+\overline{O}},$
(B1)

where we have set

$\left|\overline{U}\right|^{2}+\left|\overline{V}\right|^{2}=\cosh^{2}2g-\text{Re}\left(e^{i\phi}\overline{I}\right)\sinh^{2}2g,$
(B2)

and

$\overline{O}=2\left|\alpha\right|^{2}\left|u\right|^{2}(1-\left|\overline{I}\left(g,\phi\right)\right|^{2})+2\text{Re}\left(Z_{1}+Z_{2}\right),$
(B3)

with

$\displaystyle u$ $\displaystyle=-e^{-i\phi}\sinh^{2}g,$
$\displaystyle\chi\left(g,\phi\right)$
$\displaystyle=\frac{1}{\cosh^{2}g-e^{i2\phi}\sinh^{2}g},$
$\displaystyle\overline{I}\left(g,\phi\right)$
$\displaystyle=\chi^{2}\left(g,\phi\right)\exp\\{\left|\alpha\right|^{2}\left[\chi\left(g,\phi\right)-1\right]\\},$
$\displaystyle Z_{1}$ $\displaystyle=2(\left|\alpha\right|^{2}+\alpha^{\ast
2})u^{\ast}\overline{I}\left(g,\phi\right)$
$\displaystyle\times\left[\chi\left(g,\phi\right)-1\right]\cosh^{2}g,$
$\displaystyle Z_{2}$
$\displaystyle=\alpha^{2}u^{2}\overline{I}^{\ast}\left(g,2\phi\right)$
$\displaystyle\times[e^{-i2\phi}\chi^{\ast}\left(g,2\phi\right)-\overline{I}^{\ast}\left(g,2\phi\right)],$
(B4)

and the derivative of $\left\langle\hat{X}\right\rangle$

$\partial\left\langle\hat{X}\right\rangle/\partial\phi=2\text{Re}\left(Z_{3}Z_{4}\right),$
(B5)

where Re denotes the real part, and

$\displaystyle Z_{3}$
$\displaystyle=i\alpha^{\ast}u^{\ast}\overline{I}\left(g,\phi\right),$
$\displaystyle Z_{4}$
$\displaystyle=1+4e^{i2\phi}\left|u\right|\chi\left(g,\phi\right)$
$\displaystyle+2\left|\alpha\right|^{2}e^{i2\phi}\left|u\right|\chi^{2}\left(g,\phi\right).$
(B6)

Substituting Eqs. (B1) and (B5) into the error propagation formula in Eq. (6),
the explicit expression of the phase sensitivity can be derived theoretically.
In particular, when $\phi=0,$ we can get
$\left|\overline{U}\right|^{2}+\left|\overline{V}\right|^{2}=1$ and
$\overline{O}=0.$ Therefore, the standard deviation $\Delta\hat{X}=1.$
Moreover, utilizing the results from Eq. (B5) at the optimal phase point
$\phi=0,$ one can find the absolute value of the derivative

$\displaystyle\text{ \ \
}\left|\partial\left\langle\hat{X}\right\rangle/\partial\phi\right|$
$\displaystyle=\sqrt{N_{\alpha}}N_{OPA}\left[1+N_{OPA}\left(N_{\alpha}+2\right)\right]\sin\theta_{\alpha}.$
(B7)

Then, after taking $\theta_{\alpha}=\frac{\pi}{2}$ leading to
$\sin\theta_{\alpha}=1$, we can obtain Eq. (8).

Appendix C: The QFI in ideal case

For pure quantum system, the QFI can be calculated by Eq. (10), where the
average value of operator $A_{m}=\hat{b}^{\dagger m}\hat{b}^{m}$ are needed
(see Eq. (11)). In order to obtain the QFI, here we use the method of
characteristic function (CF). For any two-mode system, the CF is defined as

$C_{W}\left(z_{1},z_{2}\right)=\mathtt{Tr}\left[\rho_{out}D\left(z_{1}\right)D\left(z_{2}\right)\right],$
(C1)

where $D\left(z\right)=\exp\left(z\hat{a}^{\dagger}-z^{\ast}\hat{a}\right)$ is
the displacement operator and
$\rho_{out}=\left|\varphi_{out}\right\rangle\left\langle\varphi_{out}\right|$
is the density operator after the first OPA. Then the expectation value
$\bar{A}_{m}=\left\langle\hat{b}^{\dagger m}\hat{b}^{m}\right\rangle$ can be
calculated as

$\bar{A}_{m}=D_{m}C_{N}\left(0,z_{2}\right),$ (C2)

where
$C_{N}\left(0,z_{2}\right)=e^{\frac{1}{2}\left|z_{2}\right|^{2}}C_{W}\left(0,z_{2}\right)$
is the CF corresponding to normal ordering and
$D_{m}=\frac{\partial^{2m}}{\partial
z_{2}^{m}\partial\left(-z_{2}^{\ast}\right)^{m}}...|_{z_{2}=z_{2}^{\ast}=0}$
is a partial differential operator. Thus one can use Eq. (C2) to calculate the
expectation value $\bar{A}_{m}.$ For our scheme, the input state
$\left|\psi_{in}\right\rangle=\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}.$
After going through the first OPA and the phase shift, the output state is
given by
$\left|\psi_{\phi}\right\rangle=\hat{S}\left(\phi,k\right)\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$.
Here we should note that these average values of $\bar{A}_{m}$ are under the
state $\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$. Then one can
obtain

$\bar{A}_{m}=m!(\sinh^{2m}g)L_{m}(-\left|\alpha\right|^{2}),$ (C3)

where we have utilized the relation between Laguerre polynomials and two-
variable Hermit polynomials,

$L_{m}\left(xy\right)=\frac{\left(-1\right)^{m}}{m!}H_{m,m}\left(x,y\right),$
(C4)

and the generating function of $H_{m,m}\left(x,y\right)$ is

$\displaystyle\text{ \ \ }H_{m,m}\left(x,y\right)$
$\displaystyle=\frac{\partial^{2m}}{\partial s^{m}\partial
t^{m}}\left.\exp\left(-st+sx+ty\right)\right|_{s=t=0}.$ (C5)

Thus, substituting Eq. (11) and Eq. (C3) into Eq. (10), one can get the
explicit expression of the QFI, for the linear phase shift $\left(k=1\right)$
and Kerr nonlinear phase shift $\left(k=2\right),$ respectively,

$\displaystyle F_{1}$
$\displaystyle=4\left(\bar{A}_{2}+\bar{A}_{1}-\bar{A}_{1}^{2}\right),$
$\displaystyle F_{2}$ $\displaystyle=F_{1}+f,$ $\displaystyle
f=4\left[\bar{A}_{4}+6\left(\bar{A}_{3}+\bar{A}_{2}\right)-\bar{A}_{2}\left(\bar{A}_{2}+2\bar{A}_{1}\right)\right].$
(C6)

Appendix D: The Proof of Eqs. (32) and (33)

Using the completeness relation of Fock state, one can get

$\displaystyle\text{ \ \
}\eta^{\hat{n}}\hat{n}^{q}\underset{\lambda=0}{\overset{\infty}{\sum}}\left|\lambda\right\rangle\left\langle\lambda\right|$
$\displaystyle=\sum_{\lambda=0}^{\infty}\eta^{\lambda}\lambda^{q}\frac{\hat{b}^{\dagger\lambda}}{\lambda!}\left|0\right\rangle\left\langle
0\right|\hat{b}^{\lambda}$
$\displaystyle=:\underset{\lambda=0}{\overset{\infty}{\sum}}\eta^{\lambda}\lambda^{q}\frac{1}{\lambda!}\hat{b}^{\dagger\lambda}e^{-\hat{b}^{\dagger}\hat{b}}\hat{b}^{\lambda}\colon$
$\displaystyle=\colon\left.e^{-\hat{b}^{\dagger}\hat{b}}\underset{\lambda=0}{\overset{\infty}{\sum}}\frac{(\eta\hat{b}^{\dagger}\hat{b})^{\lambda}}{\lambda!}\frac{\partial^{q}}{\partial
x^{q}}e^{x\lambda}\right|_{x=0}\colon$
$\displaystyle=\colon\left.\frac{\partial^{q}}{\partial x^{q}}e^{\left(\eta
e^{x}-1\right)\hat{b}^{\dagger}\hat{b}}\right|_{x=0}\colon,$ (D1)

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

$\left|0\right\rangle\left\langle 0\right|=\colon
e^{-\hat{b}^{\dagger}\hat{b}}\colon.$ (D2)

Then, using Eq. (D1), one can calculate the following sum operator, i.e.,

$\displaystyle\underset{l=0}{\overset{\infty}{\sum}}\frac{\left(1-\eta\right)^{l}}{l!}l^{p}\hat{b}^{\dagger
l}\eta^{\hat{n}}\hat{n}^{q}\hat{b}^{l}$
$\displaystyle=\underset{l=0}{\overset{\infty}{\sum}}\frac{\left(1-\eta\right)^{l}}{l!}l^{p}\colon\left.(\hat{b}^{\dagger}\hat{b})^{l}\frac{\partial^{q}}{\partial
x^{q}}e^{\left(\eta
e^{x}-1\right)\hat{b}^{\dagger}\hat{b}}\right|_{x=0}\colon$
$\displaystyle=\colon\left.\underset{l=0}{\overset{\infty}{\sum}}\frac{\left[\left(1-\eta\right)\hat{b}^{\dagger}\hat{b}\right]^{l}}{l!}\frac{\partial^{q+p}}{\partial
x^{q}\partial y^{p}}e^{\left(\eta
e^{x}-1\right)\hat{b}^{\dagger}\hat{b}+yl}\right|_{x=y=0}\colon$
$\displaystyle=\colon\left.\frac{\partial^{q+p}}{\partial x^{q}\partial
y^{p}}e^{\left[\eta
e^{x}+\left(1-\eta\right)e^{y}-1\right]\hat{b}^{\dagger}\hat{b}}\right|_{x=y=0}\colon$
$\displaystyle=\left.\frac{\partial^{q+p}}{\partial x^{q}\partial y^{p}}[\eta
e^{x}+\left(1-\eta\right)e^{y}]^{\hat{b}^{\dagger}\hat{b}}\right|_{x=y=0}.$
(D3)

In the last step, we have used the following operator identity about
$e^{\lambda\hat{b}^{\dagger}\hat{b}}$ i.e.,

$e^{\lambda\hat{b}^{\dagger}\hat{b}}=\colon
e^{\left(e^{\lambda}-1\right)\hat{b}^{\dagger}\hat{b}}\colon,$ (D4)

to remove the symbol of normal ordering.

Appendix E: The specific expression of $C_{Q}$

Using Eqs. (27), (33) and Eqs. (29) and (31), one can get

$\displaystyle C_{Q}$
$\displaystyle=4(W_{1}^{2}\left\langle\Delta^{2}\hat{n}^{2}\right\rangle-
W_{2}\left\langle\hat{n}^{3}\right\rangle+W_{3}\left\langle\hat{n}^{2}\right\rangle$
$\displaystyle\text{ \ \ \ }-W_{4}\left\langle\hat{n}\right\rangle-
W_{5}\left\langle\hat{n}^{2}\right\rangle\left\langle\hat{n}\right\rangle-
W_{6}\left\langle\hat{n}\right\rangle^{2}),$ (E1)

where we have set

$\displaystyle W_{1}$ $\displaystyle=w_{1}\eta^{2}-2w_{2}\eta-\mu_{2},$
$\displaystyle W_{2}$
$\displaystyle=2\eta\left(3w_{1}^{2}\eta^{3}-3w_{3}\eta^{2}-w_{4}\eta+w_{5}\right),$
$\displaystyle W_{3}$
$\displaystyle=\eta\left(11w_{1}^{2}\eta^{3}-2w_{6}\eta^{2}+w_{7}\eta-4w_{1}w_{2}\right),$
$\displaystyle W_{4}$
$\displaystyle=\eta\left(6\eta^{3}-12\eta^{2}+7\eta-1\right)w_{1}^{2},$
$\displaystyle W_{5}$ $\displaystyle=2\eta\left(1-\eta\right)w_{1}W_{1},$
$\displaystyle W_{6}$
$\displaystyle=\eta^{2}\left(1-\eta\right)^{2}w_{1}^{2},$ (E2)

as well as

$\displaystyle w_{1}$ $\displaystyle=1+2\mu_{1}-\mu_{2},$ $\displaystyle
w_{2}$ $\displaystyle=\mu_{1}-\mu_{2},$ $\displaystyle w_{3}$
$\displaystyle=1+2\left(3\mu_{1}-2\mu_{2}\right)$ $\displaystyle\text{ \ \ \ \
}+\left(2\mu_{1}-\mu_{2}\right)\left(4\mu_{1}-3\mu_{2}\right),$ $\displaystyle
w_{4}$
$\displaystyle=7\mu_{2}-6\mu_{1}+24\mu_{1}\mu_{2}-14\mu_{1}^{2}-9\mu_{2}^{2},$
$\displaystyle w_{5}$ $\displaystyle=\mu_{2}w_{1}-2w_{2}^{2},$ $\displaystyle
w_{6}$
$\displaystyle=9+40\mu_{1}-22\mu_{2}+44\mu_{1}^{2}-48\mu_{1}\mu_{2}+13\mu_{2}^{2},$
$\displaystyle w_{7}$
$\displaystyle=7+40\mu_{1}-26\mu_{2}+52\mu_{1}^{2}-64\mu_{1}\mu_{2}+19\mu_{2}^{2}.$
(E3)

In particular, when $\mu_{1}=\mu_{2}=-1$, one can obtain the expected result,
i.e.,

$F_{L_{2}}\leqslant C_{Q}=4\left\langle\Delta^{2}\hat{n}^{2}\right\rangle.$
(E4)

While for $\mu_{1}=\mu_{2}=0,$ corresponding to the photon losses before the
Kerr nonlinear phase shift, one can get the upper bound, i.e.,

$\displaystyle F_{L_{2}}$ $\displaystyle\leqslant
C_{Q}=4[\eta^{4}\left\langle\Delta^{2}\hat{n}^{2}\right\rangle+6\eta^{3}\left(1-\eta\right)\left\langle\hat{n}^{3}\right\rangle$
$\displaystyle\text{ \ \ \
}+\eta^{2}\left(11\eta^{2}-18\eta+7\right)\left\langle\hat{n}^{2}\right\rangle$
$\displaystyle\text{ \ \ \
}-\eta\left(6\eta^{3}-12\eta^{2}+7\eta-1\right)\left\langle\hat{n}\right\rangle$
$\displaystyle\text{ \ \ \
}-2\eta^{3}\left(1-\eta\right)\left\langle\hat{n}^{2}\right\rangle\left\langle\hat{n}\right\rangle$
$\displaystyle\text{ \ \ \
}-\eta^{2}\left(1-\eta\right)^{2}\left\langle\hat{n}\right\rangle^{2}],$ (E5)

as expected.

In order to minimize $C_{Q},$ one can take $\partial
C_{Q}/\partial\mu_{1}=\partial C_{Q}/\partial\mu_{2}=0$ for this purpose.
Using Eqs. (E1)-(E3), it is not difficult to obtain two optimization
parameters $\mu_{1opt}$ and $\mu_{2opt}$, which are, respectively, given by

$\mu_{1opt}=\frac{BE-CD}{AD-2\eta B^{2}},\mu_{2opt}=\frac{AE-2\eta
BC}{AD-2\eta B^{2}},$ (E6)

where we have set $A=2B_{1}H,$ $B=B_{2}H,$ $C=B_{3}H,$ $D=B_{4}H,$ and $E=\eta
B_{5}H.$ Here the column matrix $H$ and row matrix
$B_{j}\left(j=1,2,3,4,5\right)$ are defined as the following

$\displaystyle H$
$\displaystyle=(\left\langle\Delta^{2}\hat{n}^{2}\right\rangle,\left\langle\hat{n}^{3}\right\rangle,\left\langle\hat{n}^{2}\right\rangle,$
$\displaystyle\text{ \ \ \
}\left\langle\hat{n}\right\rangle,\left\langle\hat{n}^{2}\right\rangle\left\langle\hat{n}\right\rangle,\left\langle\hat{n}\right\rangle^{2})^{T},$
$\displaystyle A_{1}$ $\displaystyle=\eta-1,$ $\displaystyle A_{2}$
$\displaystyle=6\eta^{2}-6\eta+1,$ $\displaystyle A_{3}$
$\displaystyle=11\eta^{2}-11\eta+2,$ $\displaystyle A_{4}$
$\displaystyle=2\eta-1,$ $\displaystyle B_{1}$ $\displaystyle=\left(\eta
A_{1},-A_{2},A_{3},-A_{2},2\eta A_{1},-\eta A_{1}\right),$ $\displaystyle
B_{2}$ $\displaystyle=(A_{1}^{2},-3A_{1}A_{4},A_{3}-A_{4},$
$\displaystyle\text{ \ \ \ }-A_{2},A_{1}A_{4},-\eta A_{1}),$ $\displaystyle
B_{3}$ $\displaystyle=\left(\eta^{2},-3\eta A_{4},A_{3}+A_{4},-A_{2},\eta
A_{4},-\eta A_{1}\right),$ $\displaystyle B_{4}$
$\displaystyle=\eta(\eta^{-1}A_{1}^{3},-6A_{1}^{2},A_{3}-2A_{4},$
$\displaystyle\text{ \ \ \ }-A_{2},2A_{1}^{2},-\eta A_{1}),$ $\displaystyle
B_{5}$ $\displaystyle=\left(\eta
A_{1},-A_{2},A_{3},-A_{2},\eta^{2}+A_{1}^{2},-\eta A_{1}\right),$ (E7)

where the average value $\left\langle\cdot\right\rangle$ is in the state after
the first OPA. Here we should note that, only $H$ are dependent on the input
state, and the other quantities (such as $A_{j},B_{j}$) are independent of the
input state. If $C_{Q}$ can take the minimum value, then it is also the QFI
$F_{L_{2}}$ of Kerr nonlinear phase shift in the presence of photon losses. In
our scheme, if we choose the states
$\left|\psi_{in}\right\rangle=\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$
as the inputs of the KSU(1,1) interferometer, then the states after the first
OPA is $\hat{S}\left(\xi_{1}\right)\left|\psi_{in}\right\rangle$. Thus the
column matrix $H$ can be calculated as

$\displaystyle H$
$\displaystyle=[\frac{F_{2}}{4},\left(\bar{A}_{3}+3\bar{A}_{2}+\bar{A}_{1}\right),$
$\displaystyle\text{ \ \ \
}\left(\bar{A}_{2}+\bar{A}_{1}\right),\bar{A}_{1},\bar{A}_{1}\left(\bar{A}_{2}+\bar{A}_{1}\right),\bar{A}_{1}^{2}]^{T},$
(E8)

where $F_{2}$ is lossless result in Eq. (12) and
$\bar{A}_{j}\left(j=1,2,3\right)$ and $W_{l}(l=1\sim 6)$ can be obtained from
Eqs. (C3) and (E2), respectively.

Further substituting Eq. (E8) and Eq. (E6) into Eq. (E1), the upper bound to
the QFI
$C_{Q}\left(\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}\right)$
in our scheme can be obtained as

$\displaystyle\text{ \ \ \
}C_{Q}\left(\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}\right)$
$\displaystyle=W_{1}^{2}F_{2}-4[W_{2}\left(\bar{A}_{3}+3\bar{A}_{2}+\bar{A}_{1}\right)$
$\displaystyle\text{ \ \ \
}-W_{3}\left(\bar{A}_{2}+\bar{A}_{1}\right)+W_{4}\bar{A}_{1}$
$\displaystyle\text{ \ \ \
}+W_{5}\bar{A}_{1}\left(\bar{A}_{2}+\bar{A}_{1}\right)+W_{6}\bar{A}_{1}^{2}],$
(E9)

which is just the analytical expression of the QFI. In $W_{l}(l=1\sim 6)$, the
variational parameters $\mu_{1}$ and $\mu_{2}$ should be replaced,
respectively, by

$\displaystyle\mu_{1opt}\left(\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}\right)$
$\displaystyle=\frac{BE-CD}{AD-2\eta B^{2}},$
$\displaystyle\mu_{2opt}\left(\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}\right)$
$\displaystyle=\frac{AE-2\eta BC}{AD-2\eta B^{2}},$ (E10)

where $A=2B_{1}H,$ $B=B_{2}H,$ $C=B_{3}H,$ $D=B_{4}H,$ and $E=\eta B_{5}H$
with the column matrix $H$ is shown in Eq. (E8).

In addition, for a comparison between the linear phase shift and the nonlinear
one (our scheme), here we give the QFI with the linear phase shift in the
presence of photon losses, for several different input states, including
$\left|\alpha\right\rangle_{a}\otimes\left|0\right\rangle_{b}$ (CS&VS),
$\left|\alpha\right\rangle_{a}\otimes\left|\beta\right\rangle_{b}$ (CS&CS) and
$\left|\alpha\right\rangle_{a}\otimes\left|\varsigma,0\right\rangle_{b}$
(CS&SVS). In a similar way to derive Eq. (E9), for these states above together
with the linear phase shift, the QFI $F_{L_{1}}$ can be calculated as 68

$\displaystyle\text{ \ \ }F_{L_{1}\left(CS\&VS\right)}$
$\displaystyle=\frac{4\eta
F_{1}\bar{A}_{1}}{\left(1-\eta\right)F_{1}+4\eta\bar{A}_{1}},$
$\displaystyle\text{ \ \ }F_{L_{1}\left(CS\&CS\right)}$
$\displaystyle=\frac{4\eta
F_{\left(CS\&CS\right)}\left\langle\hat{n}\right\rangle_{\left(CS\&CS\right)}}{\left(1-\eta\right)F_{\left(CS\&CS\right)}+4\eta\left\langle\hat{n}\right\rangle_{\left(CS\&CS\right)}},$
$\displaystyle\text{ \ \ }F_{L_{1}\left(CS\&SVS\right)}$
$\displaystyle=\frac{4\eta
F_{\left(CS\&SVS\right)}\left\langle\hat{n}\right\rangle_{\left(CS\&SVS\right)}}{\left(1-\eta\right)F_{\left(CS\&SVS\right)}+4\eta\left\langle\hat{n}\right\rangle_{\left(CS\&SVS\right)}},$
(E11)

where $\left\langle\hat{n}\right\rangle_{\left(CS\&CS\right)}$ and
$\left\langle\hat{n}\right\rangle_{\left(CS\&SVS\right)}$ are given by

$\displaystyle\text{ \ \
}\left\langle\hat{n}\right\rangle_{\left(CS\&CS\right)}$
$\displaystyle=\left(\left|\alpha\right|\sinh g+\left|\beta\right|\cosh
g\right)^{2}+\sinh^{2}g,$ $\displaystyle\text{ \ \
}\left\langle\hat{n}\right\rangle_{\left(CS\&SVS\right)}$
$\displaystyle=\left(\left|\alpha\right|^{2}+1\right)\sinh^{2}g+\cosh^{2}g\left(\sinh^{2}r\right),$
(E12)

and $F_{\left(CS\&CS\right)}$ and $F_{\left(CS\&SVS\right)}$ are the lossless
results which can be seen from Ref. 63 , i.e.,

$\displaystyle\text{ \ \ }F_{\left(CS\&CS\right)}$
$\displaystyle=(\left|\alpha\right|^{2}+\left|\beta\right|^{2})\cosh
4g+\sinh^{2}(2g)$ $\displaystyle\text{ \ \
}+2\left|\alpha\right|\left|\beta\right|\sinh
4g+\left|\alpha\right|^{2}+\left|\beta\right|^{2}$ $\displaystyle\text{ \ \
}-2\left(\left|\alpha\right|^{2}-\left|\beta\right|^{2}\right)\cosh 2g,$ (E13)
$\displaystyle\text{ \ \ }F_{\left(CS\&SVS\right)}$
$\displaystyle=\cosh^{2}\left(2g\right)[\frac{1}{2}\sinh^{2}\left(2r\right)+\left|\alpha\right|^{2}]$
$\displaystyle\text{ \ \
}+\sinh^{2}(2g)(\left|\alpha\right|^{2}e^{2r}+\cosh^{2}r)$
$\displaystyle\text{ \ \ }+\left|\alpha\right|^{2}\left(1-2\cosh 2g\right)$
$\displaystyle\text{ \ \ }+\frac{1}{4}(\cosh 4r-1)(2\cosh 2g+1).$ (E14)

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