# Quantum parameter estimation of non-Hermitian systems with optimal
measurements

Xinglei Yu  Chengjie Zhang<EMAIL_ADDRESS>School of Physical
Science and Technology, Ningbo University, Ningbo, 315211, China

###### Abstract

Quantum parameter estimation with Hermitian systems has been applied in
various fields, but there are relatively few results concerning non-Hermitian
systems. Here, we study the quantum parameter estimation for general non-
Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher
information (QFI) for pure states. Furthermore, we propose the condition for
optimal measurements, which is applicable to both Hermitian and non-Hermitian
Hamiltonians. To illustrate these results, we calculate and study the QFI of a
specific $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian, and give the
optimal measurement. Surprisingly, we find some interesting properties of this
$\mathcal{PT}$-symmetric Hamiltonian QFI, such as the mutations in QFI at EP.
Moreover, we also compare the variance of estimation generated by the optimal
measurement with the theoretical precision bound to verify the condition for
optimal measurements we proposed.

##  I Introduction

Quantum parameter estimation is a theory committed to high-precision
measurement of parameters in a quantum system, which plays a crucial role in
modern science and engineering. Cramér-Rao bound (CRB) and Fisher information
are of great importance in classical parameter estimation, which characterize
and limit the estimation precision. Helstrom rev1  and Holevo rev2  proposed
the parameter-based uncertainty relation, and pioneered quantum metrology
based on quantum parameter estimation. After that, Braunstein _et al_. rev3 ;
rev4  extended Fisher information to the quantum regime, and proposed quantum
Fisher information (QFI) which is the optimal Fisher information over
different valid quantum measurements. It is a metric that characterizes the
maximum amount of attainable information about the given parameter and
provides a new ultimate bound often called quantum CRB.

With the development of technology, precision measurement has attracted
extensive attention in various fields, which has also promoted the development
of quantum parameter estimation theory. Recently, scientists have found that
QFI played an essential role in quantum information theory, especially in
quantum entanglement theory. The QFI can be used as criteria for bipartite
entanglement rev5  and multipartite entanglement, it has been proved that only
genuine multipartite entanglement is able to reach a maximal metrological
sensitivity rev6 ; rev7 . The QFI was also discovered to be related to the
speed of quantum evolution rev8 ; rev9 ; rev10  and used for the proper
definition of a macroscopic superposition rev11 ; rev12 . A large multipartite
entanglement with large QFI can evolve faster than a state with a small QFI.
In addition, QFI has also been applied to quantum Zeno effect rev13 ; rev14 .

In recent years, quite a few studies have shown that non-Hermitian systems
have many interesting properties which attracts considerable attention. An
important discovery is that a non-Hermitian Hamiltonian with exact parity-time
(PT) symmetry may exhibit entirely real spectrum rev15 ; rev16 ; rev17 ; rev18
; rev19 . In such $\mathcal{PT}$-symmetric systems, eigenstates are not
orthogonal, and states do not conserve probabilities after evolution.
Interestingly, the $\mathcal{PT}$-symmetry may be suddenly broken once the
non-Hermiticity parameter exceeds a certain critical value rev20 ; rev21 .
These discoveries have promoted the new development in material science rev22
; rev23 , topology rev24 ; rev25  and especially optics rev26 ; rev27 ; rev28
; rev29 ; rev30 .

In previous quantum metrology researches, there have been a great quantity of
works on Hermitian systems hev1 ; hev2 ; hev3 ; hev4 ; hev5 ; hev6 ; hev7 ;
hev8 ; hev9 ; hev10 ; hev11 ; hev12 ; hev13 ; hev14 . However, only a few
quantum sensors based on non-Hermitian systems have been proposed recently
rev31 ; rev32 ; rev33 ; rev34 ; nev1 ; nev2 ; nev3 ; nev4 ; nev5 ; nev6 ; nev7
, these researches show that sensors may have enhanced sensitivity in
$\mathcal{PT}$-symmetric non-Hermitian systems. The optimal homodyne-based
measurement scheme and fundamental bounds of non-Hermitian quantum sensor has
been studied in Ref. opm1 , it considers the effect of noise as induced by
non-Hermitianity. And the parameter estimation in $\mathcal{PT}$ symmetrical
cavity magnonics system has also been studied emp1 . Recently, the quantum
Cramer-Rao bound has been also extend to the non-Hermitian regime with non-
Hermitian Symmetric Logarithmic Derivative emnh . Considering the new
developments in physics arising from the studies of non-Hermitian systems over
the past two decades, it is significant to promote the study of quantum
metrology in non-Hermitian systems.

In this paper, we derive a general intuitive expression of QFI for arbitrary
parameter-independent non-Hermitian Hamiltonians and find the condition for
optimal measurements based on the error-propagation formula. According to the
expression we proposed, we calculate and study the QFI of a specific
$\mathcal{PT}$-symmetric non-Hermitian Hamiltonian and find the optimal
measurement. Interestingly, we find that the QFI of this
$\mathcal{PT}$-symmetric Hamiltonian is discontinuous at EP, it will reduce to
zero as EP is approached, but changes suddenly when the EP is exactly reached.
We also find that the channel QFI cqfi  which is maximized after performing
the input optimization does not tend to zero near EP, and the expressions of
the channel QFI are the same in both cases of $\mathcal{PT}$-symmetry broken
and not. Comparing the estimation variance generated by the optimal
measurement with the lower bound of quantum CRB, we verify the condition for
optimal measurement that we proposed.

##  II QFI for general non-Hermitian Hamiltonians

Before we calculate the QFI for parameter-independent non-Hermitian
Hamiltonians, let us first review some basic concepts of the quantum
estimation theory. The precision of estimation is usually characterized by the
variance of the estimator, so we expect to reduce the variance as much as
possible. But how can we know the lower bound of variance? According to the
estimation theory, the variance of any unbiased estimator $\hat{\theta}$ is
limited by quantum CRB $V(\hat{\theta})\geq 1/(n\mathcal{F}_{\theta})$, where
$n$ is the number of measurements and $\theta$ is the actual parameter which
we expect to estimate, $\hat{\theta}$ is the estimator of $\theta$, and
$\mathcal{F}_{\theta}$ is the quantum Fisher information which gives the
theoretical ultimate bound of parameter estimation precision. Note that the
quantum CRB is asymptotically achievable, which requires the number of
measurements $n$ to be large. The general expression of QFI is
$\mathcal{F}_{\theta}=\mathrm{Tr}[L_{\theta}^{2}\rho_{\theta}]$, where
$L_{\theta}$ is a self-adjoint operator called Symmetric Logarithmic
Derivative (SLD), satisfies the equation
$(L_{\theta}\rho_{\theta}+\rho_{\theta}L_{\theta})/2=\partial_{\theta}\rho_{\theta}$.
For an arbitrary state written in its eigenbasis
$\rho_{\theta}=\sum_{n}c_{n}\ket{\psi_{n}}\bra{\psi_{n}}$, the expression of
QFI is rev1 ; rev2

$\displaystyle\mathcal{F}_{\theta}=2\sum_{n,m}\frac{|\bra{\psi_{n}}\partial_{\theta}\rho_{\theta}\ket{\psi_{m}}|^{2}}{c_{n}+c_{m}},$
(1)

where $c_{n}+c_{m}$ is required not to be zero. For unitary families and pure
states model, the final state is
$\rho_{\theta}=|\psi_{\theta}\rangle\langle\psi_{\theta}|=\hat{U}_{\theta}\rho_{0}\hat{U}_{\theta}^{\dagger}$
and the QFI is
$\mathcal{F}_{\theta}=4\bra{\psi_{0}}(\Delta\hat{H})^{2}\ket{\psi_{0}}$, where
$\hat{U}_{\theta}=e^{-i\hat{H}\theta}$ is an unitary operator and $\hat{H}$ is
the corresponding Hermitian generator.

Now let us discuss the QFI for parameter-independent non-Hermitian
Hamiltonians in the case of pure states. If we turn to non-Hermitian
Hamiltonians, the evolution operator is no more unitary due to
$\hat{H}\neq\hat{H}^{\dagger}$, and the final state $\ket{\psi_{\theta}}$ is
no longer normalized after evolution. Therefore, the QFI for pure states may
no longer be written as the variance of Hamiltonian. According to the Born
rule, we have $p(x|\theta)=\mathrm{Tr}[\pi_{x}\rho_{\theta}]$, this requires
$\rho_{\theta}$ to be normalized, so we cannot directly substitute
$\rho_{\theta}$ into Eq. (1) to get the result. As for measurement in non-
Hermitian systems, the probabilities of measurement outcomes always sum up to
$1$, even if there are gains and losses during evolution process. For
instance, the number of photons is reduced from $N_{0}$ to
$N^{\prime}=N_{0}\langle\psi_{\theta}|\psi_{\theta}\rangle$ after non-unitary
evolution, the final states are measured with a set of projection operator
$\Pi=\sum_{i}|i\rangle\langle i|$, then the total probability is
$\sum_{i}P_{i}=\sum_{i}N_{i}/N^{\prime}=(N_{0}\sum_{i}\langle\psi_{\theta}|i\rangle\langle
i|\psi_{\theta}\rangle)/(N_{0}\langle\psi_{\theta}|\psi_{\theta}\rangle)=1$,
where $N_{i}$ represents the number of photons whose measurement outcome is
$|i\rangle$. Thus, we assume that the pure state $\tilde{\rho}_{\theta}$ is
the normalized final state :

$\displaystyle\tilde{\rho}_{\theta}=|\varphi_{\theta}\rangle\langle\varphi_{\theta}|=\frac{\rho_{\theta}}{K_{\theta}},$
(2)

where $K_{\theta}$ is the normalization coefficient, and
$\ket{\varphi_{\theta}}=(\hat{U}_{\theta}/\sqrt{K_{\theta}})\ket{\psi_{0}}$.
Note $K_{\theta}$ is the inner product of $\ket{\psi_{\theta}}$ itself, it
must be non-negative. According to Eq. (1), for the normalized
$\tilde{\rho}_{\theta}$, we derive the QFI as follows,

$\displaystyle\mathcal{F}_{\theta}=4(\langle\hat{H}^{\dagger}\hat{H}\rangle_{\theta}-\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}).$
(3)

Here, we define $\langle\hat{M}\rangle_{\theta}$ as the expectation value of
the operator $\hat{M}$ on the normalized final state $\ket{\varphi_{\theta}}$
($\langle\hat{M}\rangle_{\theta}=\bra{\varphi_{\theta}}\hat{M}\ket{\varphi_{\theta}}$).
The detailed derivation can be found in Appendix A. This expression is valid
under the assumption that the evolution of non-Hermitian system satisfies the
Schrödinger equation: $i\partial_{t}\ket{\psi}=\hat{H}\ket{\psi}$. For open
systems, if the non-Hermitian Hamiltonian can be regard as the effective
description of the open system qtj , our result is also valid. This result is
quite similar to QFI of Hermitian Hamiltonian, which is four times the
variance of non-Hermitian generator, but here, the QFI depends on the
normalized final state $\tilde{\rho}_{\theta}$. Obviously, the QFI is the
parameter-dependent function, we can infer that for given non-Hermitian
Hamiltonians and initial states, the QFI is no more a constant, but changes
with the parameter $\theta$.

Note that the QFI we proposed above is based on a successful detection event,
it does not represent the ultimate estimation precision for given resource of
probe states. For example, if we input $N_{0}$ probe states, the total number
of final states that we are able to detect is $N^{\prime}=K_{\theta}N_{0}$.
According to the quantum Cramér-Rao bound, the ultimate estimation precision
is $V(\hat{\theta})\geq 1/(N^{\prime}\mathcal{F}_{\theta})$. Therefore, for
given input resource, we could define
$\mathcal{I}_{\theta}=K_{\theta}\mathcal{F}_{\theta}$ to characterize to
estimation precision, it indicates that the gain and loss during evolution
will effect the estimation precision. As for the specific implementation of
non-Hermitian system, the evolution could be a little different from the real
theoretical non-Hermitian system evolution, e.g., the effective evolution
could be $\hat{U}^{\prime}_{\theta}=f(\theta)\hat{U}_{\theta}$ spn1 ; spn2 ,
where $f(\theta)$ is a function. To characterize the estimation precision of
effevtive implementations of non-Hermitian systems, we just need to replace
$\hat{U}_{\theta}$ with $\hat{U}^{\prime}_{\theta}$ to calculate the
normalization coefficient.

##  III Condition for optimal measurements

The quantum CRB and QFI give the achievable ultimate bound on precision of
parameter estimation. However, how can we actually reach this bound in
parameter-independent non-Hermitian systems? It is well-known that the optimal
measurement is obtained if we measure in the eigenbasis of SLD, and the
concise expression for SLD
$L_{\theta}=2(|\partial_{\theta}\varphi_{\theta}\rangle\langle\varphi_{\theta}|+|\varphi_{\theta}\rangle\langle\partial_{\theta}\varphi_{\theta}|)$
has been proposed already, it is also valid for non-Hermitian Hamiltonians.
However, SLD is not unique, i.e., the optimal measurement is not unique, the
above expression cannot find all SLD. Here, we propose an alternative
condition which also has a wide scope of application, and we prove that
$L_{\theta}=2(|\partial_{\theta}\varphi_{\theta}\rangle\langle\varphi_{\theta}|+|\varphi_{\theta}\rangle\langle\partial_{\theta}\varphi_{\theta}|)$
satisfies the condition and find a new SLD in Appendix B. And the condition
for optimal measurements also indicates the connection between optimal
measurements and uncertain relation. We first define
$\delta\hat{M}=\hat{M}-\langle\hat{M}\rangle_{\theta}$ and
$(\Delta\hat{M})^{2}=\langle\delta\hat{M^{\dagger}}\delta\hat{M}\rangle_{\theta}=\langle\hat{M}^{\dagger}\hat{M}\rangle_{\theta}-\langle\hat{M}^{\dagger}\rangle_{\theta}\langle\hat{M}\rangle_{\theta}$.
For an observable $\hat{A}$, the precision of estimation can be characterized
with the error-propagation formula ero1 ; ero2 ,

$\displaystyle(\Delta\theta)^{2}=\frac{(\Delta\hat{A})^{2}}{n|\partial_{\theta}\langle\hat{A}\rangle_{\theta}|^{2}}.$
(4)

Similarly, the error-propagation formula is also asymptotically right when $n$
is large. According to the Non-Hermitian uncertainty relationship
$(\Delta\hat{A})^{2}(\Delta\hat{B})^{2}\geq|\langle\hat{A}^{\dagger}\hat{B}\rangle-\langle\hat{A}^{\dagger}\rangle\langle\hat{B}\rangle|$
ur1 ; ur2 ; ur3 ; rev35 ; zxev35  which was first proposed by Pati in Ref. ur1
, we can obtain an inequality as follows,

$\displaystyle(\Delta\theta)^{2}=\frac{(\Delta\hat{A})^{2}}{n|\partial_{\theta}\langle\hat{A}\rangle_{\theta}|^{2}}\geq\frac{(\Delta\hat{A})^{2}}{4n(\Delta\hat{A})^{2}(\Delta\hat{H})^{2}}=\frac{1}{n\mathcal{F}_{\theta}},$
(5)

the right term is exactly the lower bound of quantum CRB. To saturate this
inequality, the Hermitian operator $\hat{A}$ must obey the relationship as
follows,

$\displaystyle\ket{f}=iC\ket{g},$ (6)

where $\ket{f}=\delta\hat{H}\ket{\varphi_{\theta}}$,
$\ket{g}=\delta\hat{A}\ket{\varphi_{\theta}}$ and $C$ is a real number. Notice
that Eq. (6) ensures the saturation of uncertain relationships of $\hat{A}$
and $\hat{H}$, which indicates that the saturation of uncertain relationships
is an underlying requirement to achieve quantum CRB. Except for the general
Non-Hermitian Hamiltonians, this relationship also applies to the Hermitian
Hamiltonians, which has already been obtained in Ref. rev36 . The detailed
derivation of this relationship can be found in the Appendix B. Note that this
relationship not only depends on the measurements, but is also related to the
initial states. For a given Hamiltonian, on the basis of finding the optimal
initial states to maximize the QFI, we can further find the optimal
measurements according to Eq. (6). In this way, we can estimate the parameters
with highest precision.

##  IV Example of a $\mathcal{PT}$-symmetric Hamiltonian

Let us now consider a two-level $\mathcal{PT}$-symmetric Hamiltonian to
illustrate the results in previous sections. We calculate the QFI in both
cases of $\mathcal{PT}$-symmetry broken and not, the details can be found in
the Appendix C. The $\mathcal{PT}$-symmetric Hamiltonian is rev17

$\displaystyle\hat{H}_{s}=\begin{pmatrix}re^{i\omega}&s\\\
s&re^{-i\omega}\end{pmatrix},$ (7)

where $s>0$, $r$ and $\omega$ are real. We can see that the eigenvalues
$\lambda_{\pm}=\mu\pm\sqrt{\nu_{0}}=r\cos{\omega}\pm\sqrt{s^{2}-r^{2}\sin^{2}{\omega}}$
are still real when the $\mathcal{PT}$-symmetry is not broken, i.e.,
$s^{2}>r^{2}\sin^{2}{\omega}$. Note that at EP, i.e.,
$s^{2}=r^{2}\sin^{2}{\omega}$, the eigenvalues $\lambda=r\cos\omega$ are
degenerate, and when $s^{2}<r^{2}\sin^{2}{\omega}$, $\mathcal{PT}$-symmetry
will spontaneously be broken, the two eigenvalues $\varepsilon_{\pm}=\mu\pm
i\sqrt{\nu_{1}}=r\cos{\omega}\pm i\sqrt{r^{2}\sin^{2}{\omega}-s^{2}}$ are
complex.

First we calculate the QFI of Hamiltonian $\hat{H}_{s}$ when
$\mathcal{PT}$-symmetry is not broken. Assume that the eigenstates are
normalized, we have

$\displaystyle\ket{\lambda_{+}}=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\alpha/2}\\\
e^{-i\alpha/2}\end{pmatrix},\quad\ket{\lambda_{-}}=\frac{i}{\sqrt{2}}\begin{pmatrix}e^{-i\alpha/2}\\\
-e^{i\alpha/2}\end{pmatrix},$

where $\sin{\alpha}=(r/s)\sin{\omega}$, and we set the initial state to be
normalized and arbitrary

$\displaystyle\ket{\psi_{0}}=N(\ket{\lambda_{+}}+me^{i\varphi}\ket{\lambda_{-}}),$

where $m$ and relative phase $\varphi$ are real, $N$ is the normalization
coefficient. Notice that the eigenstates are not orthogonal, we have
$\langle\lambda_{+}|\lambda_{-}\rangle=\sin\alpha$. We apply Eq. (3) and
derive the QFI as follows:

$\displaystyle\mathcal{F}_{\theta}^{(r)}=\frac{16m^{2}\nu_{0}^{2}}{[(1+m^{2})s+2mr\sin\omega\cos(2\sqrt{\nu_{0}}\theta+\varphi)]^{2}}.$
(8)

The expression shows that QFI will oscillate with $\theta$ due to
$\cos(2\sqrt{\nu_{0}}\theta+\varphi)$.

Obviously, the value of QFI would be affected by the initial states. We
further find out the optimal initial state that maximizes the QFI, which is
also called channel QFI. Notice the form of Eq. (8), we can see that the
relative phase $\varphi$ is only contained in
$\cos(2\sqrt{\nu_{0}}\theta+\varphi)$. Thus, $\varphi$ has nothing to do with
the amplitude and frequency of $\mathcal{F}_{\theta}^{(r)}$. Then we solve
$\partial\mathcal{F}_{\theta}^{(r)}/\partial m=0$, and obtain that the
extremum of $\mathcal{F}_{\theta}^{(r)}$ reaches the maximum when $m=\pm 1$.
Now we can make sure that in the case of PT-symmetry not broken, the optimal
initial state is

$\displaystyle\ket{\psi_{0}}=N(\ket{\lambda_{+}}\pm
e^{i\varphi}\ket{\lambda_{-}}).$ (9)

Figure 1: The evolution of QFI and $\mathcal{I}_{\theta}$ as functions of
$\theta$. (a) The blue solid curve corresponds to the QFI when
$\mathcal{PT}$-symmetry is not broken, we set $s=1,r=0.25,\omega=\pi/2$, and
the initial state is $\ket{\psi_{0}}=N(\ket{\lambda_{+}}+\ket{\lambda_{-}})$.
The red dashed curve corresponds to the QFI in the case of
$\mathcal{PT}$-symmetry broken, we set $s=0.25,r=1,\omega=\pi/2$, and the
initial state is
$\ket{\psi_{0}}=N(\ket{\varepsilon_{+}}-\ket{\varepsilon_{-}})$. Thus, these
two systems have the same modulus of energy, but $\sin\alpha=1/\kappa$.
According to Eq. (12), their maximum value of QFI both are 6.25. (b) The green
solid curve corresponds to the QFI when $\mathcal{PT}$-symmetry is not broken,
and the orange dashed curve corresponds to the QFI in the case of
$\mathcal{PT}$-symmetry broken. The parameters and initial states in both
cases are the same as (a).

Next we are going to calculate the QFI of $\hat{H}_{s}$ when
$\mathcal{PT}$-symmetry is broken. In this case, the normalized eigenstates
are

$\displaystyle\ket{\varepsilon_{\pm}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(r\sin\omega\pm\sqrt{\nu_{1}})^{2}+s^{2}}}\begin{pmatrix}i(r\sin\omega\pm\sqrt{\nu_{1}})\\\
s\end{pmatrix},$

and the arbitrary initial state is
$\ket{\psi_{0}}=N(\ket{\varepsilon_{+}}+me^{i\varphi}\ket{\varepsilon_{-}})$.
Notice that in this case, $(r/s)\sin{\omega}>1$, we cannot define $\alpha$ as
before, here we use $\kappa=(r/s)\sin{\omega}$. According to Eq. (3), the
corresponding QFI is

$\displaystyle\mathcal{F}_{\theta}^{(i)}=\frac{16m^{2}\nu_{1}^{2}e^{4\sqrt{\nu_{1}}\theta}}{[A(e^{4\sqrt{\nu_{1}}\theta}+m^{2})+2mse^{2\sqrt{\nu_{1}}\theta}\cos\varphi]^{2}},$
(10)

where $A=|r\sin\omega|$. We also find the optimal initial state by solving
$\partial\mathcal{F}_{\theta}^{(i)}/\partial\varphi=0$ and
$\partial\mathcal{F}_{\theta}^{(i)}/\partial m=0$. The optimal initial state
is

$\displaystyle\ket{\psi_{0}}=N(\ket{\varepsilon_{+}}+m\ket{\varepsilon_{-}}),$
(11)

where $m<0$ and $\varphi=0$. We can see that the Eq. (10) is quite different
from the Eq. (8), without the oscillation term but gain and attenuation term.
As shown in Fig. 1 (a), QFI oscillates with $\theta$ in the case of
$\mathcal{PT}$-symmetry not broken; when $\mathcal{PT}$-symmetry is broken,
QFI has only one peak and quickly attenuates with $\theta$. Therefore, the QFI
performance is better when the $\mathcal{PT}$-symmetry is not broken, and we
do more research on this. We also find that in the case of
$\mathcal{PT}$-symmetry not broken, the curve of QFI shifts along the
$\theta$-axis as we change the relative phase $\varphi$ in initial states
$\ket{\psi_{0}}$. So we can maximize the QFI for a specific parameter
$\theta_{0}$ by adjusting the relative phase $\varphi$ in initial states.
Moreover, we also plot $\mathcal{I}_{\theta}$ in both cases of
$\mathcal{PT}$-symmetry broken and not broken. As shown in Fig. 1 (b), the
amplitude of $\mathcal{I}_{\theta}$ is smaller than the QFI when
$\mathcal{PT}$-symmetry is not broken, but the maximum value of
$\mathcal{I}_{\theta}$ does not change when $\mathcal{PT}$-symmetry is broken.
That is due to the introduction of $K_{\theta}$ which make the relative phase
$\varphi$ also be able to effect the amplitude, as shown in Fig. 2, the
maximum value of $I_{\theta}$ can also reach 6.25 when $\varphi=\pi$.

Figure 2: The evolution of $\mathcal{I}_{\theta}$ in the case of
$\mathcal{PT}$-symmetry not broken as a function of $\theta$. We set that
$s=1$, $r=0.4$, $\omega=\pi/2$, $m=1$ and $\varphi$ changes from $0$ to $\pi$,
where blue dashed line ($\varphi=0$), orange dot-dashed line
($\varphi=0.5\pi$) and green solid line ($\varphi=\pi$).

With the optimal initial states, we calculate the channel QFI in both cases of
$\mathcal{PT}$-symmetry broken and not. Interestingly, we find that the
channel QFI $\mathcal{F}_{\theta,max}$ has the same expressions in both cases,

$\displaystyle\mathcal{F}_{\theta,max}=4(s+|r\sin\omega|)^{2},\quad$
$\displaystyle(s^{2}\neq r^{2}\sin^{2}\omega).$ (12)

As for EP, the eigenvalues are degenerate $\lambda_{\pm}=\mu$, if the initial
states can be expressed with eigenstates, we have

$\displaystyle\mathcal{F}_{\theta}$ $\displaystyle=$ $\displaystyle
4(\bra{\psi_{0}}e^{i\mu\theta}\mu^{2}e^{-i\mu\theta}\ket{\psi_{0}}-\bra{\psi_{0}}e^{i\mu\theta}\mu
e^{-i\mu\theta}\ket{\psi_{0}}^{2})$ $\displaystyle=$ $\displaystyle
4(\mu^{2}-\mu^{2})=0.$

However, the eigenstates also coalesce at EP, they are not a set of complete
basis, an arbitrary initial state $\ket{\psi_{0}}$ may not be able to be
expressed by the superpositions of eigenstates, so the QFI may not be zero at
EP. Based on Eq. (8) and Eq. (10), we can see that the QFI always tends to
zero as EP is approached in both cases of $\mathcal{PT}$-symmetry broken and
not, since $\nu_{0}$ and $\nu_{1}$ both tend to zero near EP. Therefore, the
QFI could be discontinuous at EP for the initial states that are linearly
independent of the eigenstates. As shown in Fig. 3(a), we set the initial
state as $|0\rangle\langle 0|$ at EP ($s=2$) which is linearly independent of
the eigenstate, the QFI $\mathcal{F}_{\theta}$ tends to zero as EP is
approached, but it suddenly increases to $16$ at EP.

Figure 3: The evolutions of QFI as a function of $s$. (a) We set
$r=2,\omega=\pi/2,m=1,\varphi=0$ and $\theta=0$, hence the system reaches EP
when $s=2$. And the initial state is $|0\rangle\langle 0|$ at EP, we have
$\mathcal{F}_{\theta}=16$. (b) We set $r=2$ and $\omega=\pi/2$, the system
reaches EP when $s=2$.

Notice that although the QFI is continuous at EP for the initial states that
can be expressed by eigenstates, the channel QFI is still discontinuous at EP.
As mentioned above, the QFI tends to zero as EP is approached, but according
to Eq. (12), the channel QFI $\mathcal{F}_{\theta,max}$ does not reduce to
zero near EP. As shown in Fig. 3(b), in both cases of $\mathcal{PT}-$symmetry
broken and not, $\mathcal{F}_{\theta,max}$ does not reduce to zero as EP
($s=2$) is approached, i.e., there still exists some intervals of $\theta$
that $\mathcal{F}_{\theta}\neq 0$ near EP. When the system exactly reaches EP,
$\mathcal{F}_{\theta,max}$ suddenly reduce to zero. We can see that in Eq. (8)
and Eq. (10), the numerator contains $\nu_{0}$ or $\nu_{1}$ which gradually
tends to zero as EP is approached, hence there is a continuous decrease in
$\mathcal{F}_{\theta}$ to zero near EP. However, for
$\mathcal{F}_{\theta,max}$, when $\nu_{0}$ and $\nu_{1}$ in the numerator tend
to zero, there is also an infinitesimal of the same order in the denominator
that tends to zero, which makes $\mathcal{F}_{\theta,max}$ not reduce to zero
near EP, more details can be found in Appendix C.

On the basis of finding the optimal initial state, we further find an optimal
measurement for Hamiltonian $\hat{H}_{s}$, for simplicity, we set
$\omega=\pi/2$. According to Eq. (6), for optimal initial state
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+e^{i\varphi}\ket{\lambda_{-}})$, the
corresponding optimal measurement is $|0\rangle\langle 0|$, where
$\ket{0}=(1,0)^{T}$, detailed calculation can be found in the Appendix D.
Moreover, to verify the condition for optimal measurement that we proposed, we
compare the estimation variance $(\Delta\theta)^{2}$ generated by
$|0\rangle\langle 0|$ with the reciprocal of QFI
$1/\mathcal{F}_{\theta}^{(r)}$. To make the expressions concise, we use
$\kappa=r/s$ here, according to error-propagation formula Eq. (4), we can
obtain the precision of the estimation for arbitrary initial state as follows,

$\displaystyle(\Delta\theta)^{2}=p/q,$ (13)

where $p=[1+m^{2}+2m\kappa\sin(2\gamma_{0}+\beta)]^{2}[1+m^{2}-2m\cos
2\gamma_{0}+4m\kappa\sin(2\gamma_{0}+\beta)](1+m^{2}+2m\cos 2\gamma_{0})$,
$q=16m^{2}\nu_{0}(1-\kappa^{2})[2m\kappa+(1+m^{2})\sin(2\gamma_{0}+\beta)]^{2}$,
$\gamma_{0}=\sqrt{\nu_{0}}\theta+\varphi/2$ and $\beta$ is determined by
$\kappa$ ($\sin\beta=\kappa,\cos\beta=-\sqrt{1-\kappa^{2}}$). In Fig. 4(a), we
can see that only if $m=1$, $(\Delta\theta)^{2}$ (the red solid curve) and the
$1/\mathcal{F}_{\theta}^{(r)}$ (blue dashed curve) overlap, i.e., the
precision of the estimation reaches the quantum CRB. In Fig .4(b), (c) and
(d), the initial state is changed, $(\Delta\theta)^{2}$ is lager than
$1/\mathcal{F}_{\theta}^{(r)}$ in some intervals, the precision of the
estimation declines. Thus, $|0\rangle\langle 0|$ is exactly the optimal
measurement of $\hat{H}_{s}$ for the optimal initial state.

Figure 4: The evolution of the variance $(\Delta\theta)^{2}$ (red solid line)
and the reciprocal of QFI $1/\mathcal{F}_{\theta}^{(r)}$ (blue dashed line) as
functions of $\theta$, we set the Hamiltonian and the relative phase to remain
unchanged, $s=0.5,r=0.25,\varphi=0$. (a) We set the initial state is optimal
$m=1.0$. The actual precision $(\Delta\theta)^{2}$ is equal to
$1/\mathcal{F}_{\theta}^{(r)}$, which means the quantum CRB is saturated. (b)
We set $m=1.1$, $(\Delta\theta)^{2}$ is larger than
$1/\mathcal{F}_{\theta}^{(r)}$, the precision of the estimation declines. In
(c) and (d), we set $m=1.2$ and $m=1.3$ respectively, the precision of the
estimation further declines.

##  V Discussions and Conclusions

We simply discuss the method to simulate the $\mathcal{PT}$-symmetric
Hamiltonian $\hat{H}_{s}$ in this section. In short, the expectation value of
$\hat{H}_{s}$ on $\ket{\varphi_{\theta}}$ can be described with the weak value
of the dilated Hamiltonian $\tilde{H}$ on the preselected state
$\ket{\tilde{\varphi}_{i}}$ and postselected state
$\ket{\tilde{\varphi}_{f}}$, i.e.
$\bra{\varphi_{\theta}}\hat{H}_{s}\ket{\varphi_{\theta}}_{\eta}/\langle\varphi_{\theta}|\varphi_{\theta}\rangle_{\eta}=\bra{\tilde{\varphi}_{i}}\tilde{H}\ket{\tilde{\varphi}_{f}}/\langle\tilde{\varphi}_{i}|\tilde{\varphi}_{f}\rangle$
rev37 , where $\eta$ is a metric operator, $\ket{\tilde{\varphi}_{i}}$ and
$\ket{\tilde{\varphi}_{f}}$ are specially dilated from
$\ket{\varphi_{\theta}}$. The right term is exactly the weak value
$\langle\tilde{H}\rangle_{w}$ of dilated Hamiltonian $\tilde{H}$, we can
obtain it through weak measurement of $\tilde{H}$. To realize the weak
measurement, we need a pointer $\hat{P}$ weakly coupling with $\tilde{H}$, the
interaction Hamiltonian can be written as
$\hat{H}_{int}=\tilde{H}\otimes\hat{P}$. It has been proved that if we choose
$\hat{P}$ as Pauli operator $\hat{\sigma}_{z}$ and the initial pointer state
is $\ket{P_{i}}=(\ket{+}+\ket{-})/\sqrt{2}$, the real and imaginary parts of
the weak value $\langle\tilde{H}\rangle_{w}$ can be read out by measuring the
expectations of Pauli matrices $\hat{\sigma}_{y}$ and $\hat{\sigma}_{z}$
respectively on postselected pointer state $\ket{P_{f}}$ nev1 .

In conclusion, we have proposed a expression of QFI with general parameter-
independent non-Hermitian Hamiltonians. It is four times the variance of non-
Hermitian generator on normalized final states, which is quite similar to the
QFI of Hermitian Hamiltonians. We also find the condition for optimal
measurements, it applies to both Hermitian and non-Hermitian Hamiltonians.
Consider a specific $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian as an
example, we calculate the QFI and find the optimal measurement. Surprisingly,
we find some interesting properties of the QFI of this
$\mathcal{PT}$-symmetric Hamiltonian. Considering the application of QFI and
$\mathcal{PT}$-symmetry in optics and quantum information, non-Hermitian
Hamiltonian QFI is expected to promote the further development of quantum
metrology and non-Hermitian physics.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
(Grant No. 11734015).

## APPENDIX A: Detailed derivation of QFI for non-Hermitian Hamiltonians

To clearly generalize the QFI to non-Hermitian systems, we start from
classical Fisher information. It is well-known that the Fisher information can
be written as

$\displaystyle F_{\theta}=\int dx\frac{1}{p(x|\theta)}\Big{[}\frac{\partial
p(x|\theta)}{\partial\theta}\Big{]}^{2}$ (A1)

where $p(x|\theta)=\mathrm{Tr}[\Pi_{x}\rho_{\theta}]$,
$\rho_{\theta}=e^{-i\hat{H}\theta}\rho_{0}e^{i\hat{H}\theta}$ is the final
state and $\\{\Pi_{x}\\}$ are the elements of a set of positive operator-value
measurement (POVM). In non-Hermitian systems, the evolution is non-unitary, as
we discussed in the main text, we need to normalized the final state. Thus, we
assume that $\tilde{\rho}_{\theta}$ is the normalized final state:

$\displaystyle\tilde{\rho}_{\theta}=\frac{e^{-iH\theta}\rho_{0}e^{i\hat{H}^{\dagger}\theta}}{\mathrm{Tr}[e^{-iH\theta}\rho_{0}e^{i\hat{H}^{\dagger}\theta}]}=\frac{\rho_{\theta}}{K_{\theta}},$
(A2)

then we have $p(x|\theta)=\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]$. Noticed
that even if the state $\rho_{0}$ evolves in non-Hermitian system, the final
state $\tilde{\rho}_{\theta}$ is still Hermitian, and obviously we can still
find a Hermitian Symmetric Logarithmic Derivative operator $L_{\theta}$
satisfies the equation
$(L_{\theta}\tilde{\rho}_{\theta}+\tilde{\rho}_{\theta}L_{\theta})/2=\partial_{\theta}\tilde{\rho}_{\theta}$,
and in the eigenbasis of
$\tilde{\rho}_{\theta}=\sum_{n}c_{n}|\psi_{n}\rangle\langle\psi_{n}|$,
$L_{\theta}$ can be written as follows

$\displaystyle
L_{\theta}=2\sum_{n,m}\frac{\bra{\psi_{n}}\partial_{\theta}\tilde{\rho}_{\theta}\ket{\psi_{m}}}{c_{n}+c_{m}}|\psi_{n}\rangle\langle\psi_{m}|,$
(A3)

where $c_{n}+c_{m}$ is required not to be zero. Obviously this expression
satisfies the equation
$(L_{\theta}\tilde{\rho}_{\theta}+\tilde{\rho}_{\theta}L_{\theta})/2=\partial_{\theta}\tilde{\rho}_{\theta}$.
Then we have

$\displaystyle\partial_{\theta}p(x|\theta)$ $\displaystyle=$
$\displaystyle\partial_{\theta}\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]=\mathrm{Tr}[\Pi_{x}\partial_{\theta}\tilde{\rho}_{\theta}]$
(A4) $\displaystyle=$
$\displaystyle\frac{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]+\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}L_{\theta}]}{2}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]+\mathrm{Tr}[(\Pi_{x}\tilde{\rho}_{\theta}L_{\theta})^{\dagger}]^{\ast}}{2}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]+\mathrm{Tr}[L_{\theta}\tilde{\rho}_{\theta}\Pi_{x}]^{\ast}}{2}$
$\displaystyle=$
$\displaystyle\mathrm{Re}\\{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]\\}.$

With this result, the Fisher information can be further written as

$\displaystyle F_{\theta}=\int
dx\frac{\mathrm{Re}\\{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]\\}^{2}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}.$
(A5)

As it known to us, the quantum Fisher information is the maximum Fisher
information over different valid quantum measurements, so we amplify the
Fisher information as

$\displaystyle F_{\theta}$ $\displaystyle=$ $\displaystyle\int
dx\frac{\mathrm{Re}\\{\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]\\}^{2}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}\leq\int
dx\frac{\big{|}\mathrm{Tr}[\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}]\big{|}^{2}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}$
(A6) $\displaystyle=$ $\displaystyle\int
dx\frac{\Big{|}\mathrm{Tr}\big{[}(\sqrt{\tilde{\rho_{\theta}}}\sqrt{\Pi_{x}})(\sqrt{\Pi_{x}}L_{\theta}\sqrt{\tilde{\rho}_{\theta}})\big{]}\Big{|}^{2}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}.$

According to the Schwartz inequality:
$|\mathrm{Tr}[AB]|^{2}\leq\mathrm{Tr}[A^{\dagger}A]\mathrm{Tr}[B^{\dagger}B]$,
we have

$\displaystyle F_{\theta}$ $\displaystyle\leq$ $\displaystyle\int
dx\frac{\Big{|}\mathrm{Tr}\big{[}(\sqrt{\tilde{\rho}_{\theta}}\sqrt{\Pi_{x}})(\sqrt{\Pi_{x}}L_{\theta}\sqrt{\tilde{\rho}_{\theta}})\big{]}\Big{|}^{2}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}$
(A7) $\displaystyle\leq$ $\displaystyle\int
dx\frac{\mathrm{Tr}\big{[}\tilde{\rho}_{\theta}\Pi_{x}\big{]}\mathrm{Tr}\big{[}\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}L_{\theta}\big{]}}{\mathrm{Tr}[\Pi_{x}\tilde{\rho}_{\theta}]}$
$\displaystyle=$ $\displaystyle\int
dx\mathrm{Tr}\big{[}\Pi_{x}L_{\theta}\tilde{\rho}_{\theta}L_{\theta}\big{]}$
$\displaystyle=$ $\displaystyle\mathrm{Tr}\big{[}(\int
dx\Pi_{x})L_{\theta}\tilde{\rho}_{\theta}L_{\theta}\big{]}=\mathrm{Tr}\big{[}\tilde{\rho}_{\theta}L_{\theta}^{2}\big{]}.$

According to the above chain of inequalities, we obtain the quantum Fisher
information

$\displaystyle\mathcal{F}_{\theta}=\mathrm{Tr}\big{[}\tilde{\rho}_{\theta}L_{\theta}^{2}\big{]}=\mathrm{Tr}\big{[}(\partial_{\theta}\tilde{\rho}_{\theta})L_{\theta}\big{]},$
(A8)

which is the upper bound of Fisher information. We can see that it is the same
as the form in Hermitian systems. Substitute Eq. (A3) into the last expression
of Eq. (A8), we obtain

$\displaystyle\mathcal{F}_{\theta}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\big{[}\partial_{\theta}\tilde{\rho}_{\theta}(2\sum_{n,m}\frac{\bra{\psi_{n}}\partial_{\theta}\tilde{\rho}_{\theta}\ket{\psi_{m}}}{c_{n}+c_{m}}|\psi_{n}\rangle\langle\psi_{m}|)\big{]}$
(A9) $\displaystyle=$ $\displaystyle
2\sum_{n,m}\frac{\bra{\psi_{n}}\partial_{\theta}\tilde{\rho}_{\theta}\ket{\psi_{m}}}{c_{n}+c_{m}}\mathrm{Tr}\big{[}\partial_{\theta}\tilde{\rho}_{\theta}|\psi_{n}\rangle\langle\psi_{m}|\big{]}$
$\displaystyle=$ $\displaystyle
2\sum_{n,m}\frac{|\bra{\psi_{n}}\partial_{\theta}\tilde{\rho}_{\theta}\ket{\psi_{m}}|^{2}}{c_{n}+c_{m}}.$

For a generic family of pure states, the normalized final state is

$\displaystyle\tilde{\rho}_{\theta}=|\varphi_{\theta}\rangle\langle\varphi_{\theta}|=\frac{|\psi_{\theta}\rangle\langle\psi_{\theta}|}{K_{\theta}}.$
(A10)

where $K_{\theta}=\langle\psi_{\theta}|\psi_{\theta}\rangle$ is the
normalization coefficient. According to Eq. (A9), we can obtain

$\displaystyle\mathcal{F}_{\theta}$ $\displaystyle=$ $\displaystyle
2[\frac{|\langle\varphi_{\theta}|\partial_{\theta}\varphi_{\theta}\rangle+\langle\partial_{\theta}\varphi_{\theta}|\varphi_{\theta}\rangle|^{2}}{2}+2\sum_{i}|\langle\partial_{\theta}\varphi_{\theta}|i\rangle|^{2}]$
(A11) $\displaystyle=$ $\displaystyle
2[\frac{|\partial_{\theta}(\langle\varphi_{\theta}|\varphi_{\theta}\rangle)|^{2}}{2}+2\sum_{i}\langle\partial_{\theta}\varphi_{\theta}|i\rangle\langle
i|\partial_{\theta}\varphi_{\theta}\rangle]$ $\displaystyle=$ $\displaystyle
4\langle\partial_{\theta}\varphi_{\theta}|(\sum_{i}|i\rangle\langle
i|)|\partial_{\theta}\varphi_{\theta}\rangle$ $\displaystyle=$ $\displaystyle
4\langle\partial_{\theta}\varphi_{\theta}|(\mathrm{I}-|\varphi_{\theta}\rangle\langle\varphi_{\theta}|)|\partial_{\theta}\varphi_{\theta}\rangle$
$\displaystyle=$ $\displaystyle
4(\langle\partial_{\theta}\varphi_{\theta}|\partial_{\theta}\varphi_{\theta}\rangle-|\langle\partial_{\theta}\varphi_{\theta}|\varphi_{\theta}\rangle|^{2}),$
$\displaystyle(L_{\theta}\rho_{\theta}+\rho_{\theta}L_{\theta})/2=\partial_{\theta}\rho_{\theta},$
(A12)

where
$\ket{\partial_{\theta}\varphi_{\theta}}=\partial_{\theta}|\varphi_{\theta}\rangle$.
And then, we further expand $|\partial_{\theta}\varphi_{\theta}\rangle$ with
$\hat{U}_{\theta}$ and $K_{\theta}$, we have
$|\partial_{\theta}\varphi_{\theta}\rangle=\partial_{\theta}(\hat{U}_{\theta}/\sqrt{K_{\theta}})|\psi_{0}\rangle$.
Thus, we can obtain the first term:

$\displaystyle\langle\partial_{\theta}\varphi_{\theta}|\partial_{\theta}\varphi_{\theta}\rangle$
$\displaystyle=$
$\displaystyle\langle\psi_{0}|\frac{\sqrt{K_{\theta}}(\partial_{\theta}\hat{U}_{\theta}^{\dagger})-\frac{(\partial_{\theta}K_{\theta})}{2\sqrt{K_{\theta}}}\hat{U}_{\theta}^{\dagger}}{K_{\theta}}$
$\displaystyle\ \ \ \ \ \
\cdot\frac{\sqrt{K_{\theta}}(\partial_{\theta}\hat{U}_{\theta})-\frac{(\partial_{\theta}K_{\theta})}{2\sqrt{K_{\theta}}}\hat{U}_{\theta}}{K_{\theta}}|\psi_{0}\rangle$
$\displaystyle=$
$\displaystyle\langle\hat{H}^{\dagger}\hat{H}\rangle_{\theta}-\frac{i(\partial_{\theta}K_{\theta})}{2K_{\theta}}(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta})+\frac{(\partial_{\theta}K_{\theta})^{2}}{4K_{\theta}^{2}},$

where $\hat{U}_{\theta}=e^{-i\hat{H}\theta}$ and
$\partial_{\theta}\hat{U}_{\theta}=-i\hat{U}_{\theta}\hat{H}$, note that
$\hat{H}$ commutes with $\hat{U}_{\theta}$. Here, we define
$\langle\hat{X}\rangle_{\theta}$ as the expectation of operator $\hat{X}$ on
normalized final state
($\langle\hat{X}\rangle_{\theta}=\bra{\varphi_{\theta}}\hat{X}\ket{\varphi_{\theta}}$).
In the same way, we can also obtain the expression for the second term as
follows:

$\displaystyle|\langle\partial_{\theta}\varphi_{\theta}|\varphi_{\theta}\rangle|^{2}$
$\displaystyle=$
$\displaystyle|\bra{\psi_{0}}\frac{\sqrt{K_{\theta}}(\partial_{\theta}\hat{U}_{\theta}^{\dagger})-\frac{(\partial_{\theta}K_{\theta})}{2\sqrt{K_{\theta}}}\hat{U}_{\theta}^{\dagger}}{K_{\theta}}\frac{\hat{U}_{\theta}}{\sqrt{K_{\theta}}}\ket{\psi_{0}}|^{2}$
$\displaystyle=$
$\displaystyle|\bra{\psi_{0}}\frac{i\hat{U}_{\theta}^{\dagger}\hat{H}^{\dagger}\hat{U}_{\theta}-\frac{(\partial_{\theta}K_{\theta})}{2K_{\theta}}\hat{U}_{\theta}^{\dagger}\hat{U}_{\theta}}{K_{\theta}}\ket{\psi_{0}}|^{2}$
$\displaystyle=$
$\displaystyle|i\langle\hat{H}^{\dagger}\rangle_{\theta}-\frac{(\partial_{\theta}K_{\theta})}{2K_{\theta}}|^{2}$
$\displaystyle=$
$\displaystyle\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}-\frac{i(\partial_{\theta}K_{\theta})}{2K_{\theta}}(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta})+\frac{(\partial_{\theta}K_{\theta})^{2}}{4K_{\theta}^{2}}.$

Substituting these two results back into Eq. (A11), we work out the concise
and intuitive expression of QFI,

$\displaystyle\mathcal{F}_{\theta}=4(\langle\hat{H}^{\dagger}\hat{H}\rangle_{\theta}-\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}),$
(A15)

If the Hamiltonian is Hermitian, this result returns to
$\mathcal{F}_{\theta}=4\bra{\psi_{0}}(\Delta\hat{H})^{2}\ket{\psi_{0}}$.

## APPENDIX B: Proof of the condition for optimal measurements

Suppose we have a Hermitian operator $\hat{A}$ as a measurement, according to
the error-propagation formula ero1 ; ero2 , we have

$\displaystyle(\Delta\theta)^{2}$ $\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\partial_{\theta}\langle\hat{A}\rangle_{\theta}|^{2}}$
(B1) $\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\bra{\partial_{\theta}\varphi_{\theta}}\hat{A}\ket{\varphi_{\theta}}+\bra{\varphi_{\theta}}\hat{A}\ket{\partial_{\theta}\varphi_{\theta}}|^{2}},$

where $\ket{\varphi_{\theta}}$ is the normalized final state.

In the case of Non-Hermitian Hamiltonians, let
$Q=\partial_{\theta}\langle\hat{A}\rangle_{\theta}=\bra{\partial_{\theta}\varphi_{\theta}}\hat{A}\ket{\varphi_{\theta}}+\bra{\varphi_{\theta}}\hat{A}\ket{\partial_{\theta}\varphi_{\theta}}$
and expand this expression, we have

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\bra{\psi_{0}}\partial_{\theta}(\frac{\hat{U}_{\theta}^{\dagger}}{\sqrt{K_{\theta}}})\hat{A}\ket{\varphi_{\theta}}+\bra{\varphi_{\theta}}\hat{A}\partial_{\theta}(\frac{\hat{U}_{\theta}}{\sqrt{K_{\theta}}})\ket{\psi_{0}}$
(B2) $\displaystyle=$
$\displaystyle\bra{\psi_{0}}\frac{\sqrt{K_{\theta}}\hat{U}_{\theta}^{\dagger}(i\hat{H}^{\dagger})-\frac{\partial_{\theta}K_{\theta}}{2\sqrt{K_{\theta}}}\hat{U}_{\theta}^{\dagger}}{K_{\theta}}\hat{A}\ket{\varphi_{\theta}}+\bar{Z}$
$\displaystyle=$
$\displaystyle\bra{\varphi_{\theta}}(i\hat{H}^{\dagger}-\frac{\partial_{\theta}K_{\theta}}{2K_{\theta}})\hat{A}\ket{\varphi_{\theta}}-\bra{\varphi_{\theta}}\hat{A}(i\hat{H}+\frac{\partial_{\theta}K_{\theta}}{2K_{\theta}})\ket{\varphi_{\theta}}$
$\displaystyle=$
$\displaystyle\bra{\varphi_{\theta}}i(\hat{H}^{\dagger}\hat{A}-\hat{A}\hat{H})\ket{\varphi_{\theta}}-\frac{\partial_{\theta}K_{\theta}}{K_{\theta}}\bra{\varphi_{\theta}}\hat{A}\ket{\varphi_{\theta}}$
$\displaystyle=$ $\displaystyle
i(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta})-\frac{\partial_{\theta}K_{\theta}}{K_{\theta}}\langle\hat{A}\rangle_{\theta},$

where
$\bar{Z}=\bra{\varphi_{\theta}}\hat{A}\partial_{\theta}(\hat{U}_{\theta}/\sqrt{K_{\theta}})\ket{\psi_{0}}$,
$\partial_{\theta}(\hat{U}_{\theta}/\sqrt{K_{\theta}})=[\sqrt{K_{\theta}}\hat{U}_{\theta}(-i\hat{H})-(\partial_{\theta}K_{\theta}/2\sqrt{K_{\theta}})\hat{U}_{\theta}]/K_{\theta}$,
and $K_{\theta}=\bra{\psi_{\theta}}\psi_{\theta}\rangle$ is the normalization
coefficient of final state, then $(\partial_{\theta}K_{\theta})/K_{\theta}$
can be written as

$\displaystyle\frac{\partial_{\theta}K_{\theta}}{K_{\theta}}$ $\displaystyle=$
$\displaystyle\frac{\partial_{\theta}(\bra{\psi_{0}}\hat{U}_{\theta}^{\dagger}\hat{U}_{\theta}\ket{\psi_{0}})}{K_{\theta}}$
(B3) $\displaystyle=$
$\displaystyle\frac{\bra{\psi_{0}}\hat{U}_{\theta}^{\dagger}i\hat{H}^{\dagger}\hat{U}_{\theta}\ket{\psi_{0}}+\bra{\psi_{0}}\hat{U}_{\theta}^{\dagger}(-i\hat{H})\hat{U}_{\theta}\ket{\psi_{0}}}{K_{\theta}}$
$\displaystyle=$
$\displaystyle\bra{\varphi_{\theta}}i(\hat{H}^{\dagger}-\hat{H})\ket{\varphi_{\theta}}$
$\displaystyle=$ $\displaystyle
i(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta}).$

Substitute the result into Eq. (B2), we have

$\displaystyle
Q=i[(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta})-(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta})\hat{A}\rangle_{\theta}].$

Then $|Q|^{2}$ can be written as

$\displaystyle|Q|^{2}$ $\displaystyle=$
$\displaystyle-(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta})^{2}-(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta})^{2}\langle\hat{A}\rangle_{\theta}^{2}$
(B5)
$\displaystyle+2(\langle\hat{H}^{\dagger}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta})(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta})\langle\hat{A}\rangle_{\theta}$
$\displaystyle=$
$\displaystyle-[(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}+\langle\hat{A}\hat{H}\rangle_{\theta})^{2}+(\langle\hat{H}^{\dagger}\rangle_{\theta}+\langle\hat{H}\rangle_{\theta})^{2}\langle\hat{A}\rangle_{\theta}^{2}$
$\displaystyle-2(\langle\hat{H}^{\dagger}\rangle_{\theta}+\langle\hat{H}\rangle_{\theta})(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}+\langle\hat{A}\hat{H}\rangle_{\theta})\langle\hat{A}\rangle_{\theta}]$
$\displaystyle+4[\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}\langle\hat{A}\hat{H}\rangle_{\theta}+\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}\langle\hat{A}\rangle_{\theta}^{2}-$
$\displaystyle\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}\langle\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta}\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{A}\rangle_{\theta}].$

According to the non-Hermitian uncertainty relationship
$(\Delta\hat{A})^{2}(\Delta\hat{B})^{2}\geq|\langle\hat{A}^{\dagger}\hat{B}\rangle-\langle\hat{A}^{\dagger}\rangle\langle\hat{B}\rangle|^{2}$
ur1 ; ur2 ; ur3 ; rev35 ; zxev35  which was first proposed by Pati in ur1 , we
have

$\displaystyle(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}$ $\displaystyle\geq$
$\displaystyle|\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{H}^{\dagger}\rangle\langle\hat{A}\rangle_{\theta}|^{2}$
$\displaystyle=$
$\displaystyle\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}\langle\hat{A}\hat{H}\rangle_{\theta}+\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}\langle\hat{A}\rangle_{\theta}^{2}$
$\displaystyle-\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}\langle\hat{A}\rangle_{\theta}-\langle\hat{A}\hat{H}\rangle_{\theta}\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{A}\rangle_{\theta},$

where
$(\Delta\hat{H})^{2}=\langle\hat{H}^{\dagger}\hat{H}\rangle_{\theta}-\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{H}\rangle_{\theta}$.
The expression in the last brackets of Eq. (B5) is exactly the same as
$(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}$. Then, we can derive an inequality as
follow,

$\displaystyle|Q|^{2}$ $\displaystyle\leq$ $\displaystyle
4(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}-[(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}+\langle\hat{A}\hat{H}\rangle_{\theta})-$
(B7)
$\displaystyle(\langle\hat{H}^{\dagger}\rangle_{\theta}+\langle\hat{H}\rangle_{\theta})\langle\hat{A}\rangle_{\theta}]^{2}.$

Here, we define an operator
$\delta\hat{M}=\hat{M}-\langle\hat{M}\rangle_{\theta}$, then we have

$\displaystyle\ket{f}=\delta\hat{H}\ket{\varphi_{\theta}}=\hat{H}\ket{\varphi_{\theta}}-\langle\hat{H}\rangle_{\theta}\ket{\varphi_{\theta}},$
(B8)
$\displaystyle\ket{g}=\delta\hat{A}\ket{\varphi_{\theta}}=\hat{A}\ket{\varphi_{\theta}}-\langle\hat{A}\rangle_{\theta}\ket{\varphi_{\theta}},$
(B9)
$\displaystyle\bra{f}g\rangle=\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{H}^{\dagger}_{\theta}\rangle\langle\hat{A}\rangle_{\theta}.$
(B10)

Then the inequality Eq. (B7) can be written as

$\displaystyle|Q|^{2}$ $\displaystyle\leq$ $\displaystyle
4(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}-[(\langle\hat{H}^{\dagger}\hat{A}\rangle_{\theta}-\langle\hat{H}^{\dagger}\rangle_{\theta}\langle\hat{A}\rangle_{\theta})+$
(B11)
$\displaystyle(\langle\hat{A}\hat{H}\rangle_{\theta}-\langle\hat{H}\rangle_{\theta}\langle\hat{A}\rangle_{\theta})]^{2}$
$\displaystyle=$ $\displaystyle
4(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}-(\bra{f}g\rangle+\bra{g}f\rangle)$
$\displaystyle\leq$ $\displaystyle 4(\Delta\hat{H})^{2}(\Delta\hat{A})^{2}.$

According to the Cauchy-Schwarz inequality, the first inequality is saturated
only if $\ket{f}=C\ket{g}$, where $C$ is a constant. Furthermore, if $C$ is an
imaginary number, the second inequality will be saturated. Then we have

$\displaystyle(\Delta\theta)^{2}$ $\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\partial_{\theta}\langle\hat{A}\rangle_{\theta}|^{2}}\geq\frac{1}{4(\Delta\hat{H})^{2}}=\frac{1}{n\mathcal{F}_{\theta}}.$
(B12)

Now we obtain the condition of optimal measurements as follows,

$\displaystyle\ket{f}=iC\ket{g},$ (B13)

where $C$ is a real number. Actually, this relationship is also useful for
Hermitian Hamiltonians, which has already been obtained in Ref. rev36 .

As for Hermitian Hamiltonians, the error-propagation formula can be expanded
as follows:

$\displaystyle(\Delta\theta)^{2}$ $\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\partial_{\theta}\langle\hat{A}\rangle_{\theta}|^{2}}$
(B14) $\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\bra{\partial_{\theta}\varphi_{\theta}}\hat{A}\ket{\varphi_{\theta}}+\bra{\varphi_{\theta}}\hat{A}\ket{\partial_{\theta}\varphi_{\theta}}|^{2}}$
$\displaystyle=$
$\displaystyle\frac{(\Delta\hat{A})^{2}}{n|\langle\hat{H}\hat{A}-\hat{A}\hat{H}\rangle|^{2}}.$

According to Heisenberg uncertainty relation, we have
$|\langle\hat{H}A-A\hat{H}\rangle|^{2}\leq
4(\Delta\hat{A})^{2}(\Delta\hat{H})^{2}$. Therefore, the variance of $\theta$
is limited,

$\displaystyle(\Delta\theta)^{2}\geq\frac{1}{4n(\Delta\hat{H})^{2}}=\frac{1}{n\mathcal{F}_{\theta}}.$
(B15)

The condition that saturates the inequality is also $\ket{f}=iC\ket{g}$.
Hofmann has proposed this condition in 2009 rev36 .

We next prove that Symmetric Logarithmic Derivative (SLD) satisfies Eq. (B13).
As we mentioned above, SLD $L_{\theta}$ satisfies the equation
$(L_{\theta}\tilde{\rho}_{\theta}+\tilde{\rho}_{\theta}L_{\theta})/2=\partial_{\theta}\tilde{\rho}_{\theta}$.
And we have

$\displaystyle\partial_{\theta}(\tilde{\rho}^{2}_{\theta})=(\partial_{\theta}\tilde{\rho}_{\theta})\tilde{\rho}_{\theta}+\tilde{\rho}_{\theta}(\partial_{\theta}\tilde{\rho}_{\theta})$
(B16)
$\displaystyle\partial_{\theta}(\tilde{\rho}^{2}_{\theta})=\partial_{\theta}(\tilde{\rho}_{\theta}).$
(B17)

Obviously, SLD is $2\partial_{\theta}\tilde{\rho}_{\theta}$ for pure state
model, i.e.,

$\displaystyle
L_{\theta}=2(|\partial_{\theta}\varphi_{\theta}\rangle\langle\varphi_{\theta}|+|\varphi_{\theta}\rangle\langle\partial_{\theta}\varphi_{\theta}|).$
(B18)

We further expand $L_{\theta}$ as follows,

$\displaystyle L_{\theta}$ $\displaystyle=$ $\displaystyle
2(|\partial_{\theta}\varphi_{\theta}\rangle\langle\varphi_{\theta}|+|\varphi_{\theta}\rangle\langle\partial_{\theta}\varphi_{\theta}|)$
$\displaystyle=$ $\displaystyle
2\Big{[}(-i\hat{H}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|+|\varphi_{\theta}\rangle\langle\varphi_{\theta}|i\hat{H}^{\dagger})-\frac{\partial_{\theta}K_{\theta}}{K_{\theta}}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|\Big{]}.$

According to Eq. (B3), we have

$\displaystyle L_{\theta}$ $\displaystyle=$
$\displaystyle-2i\Big{[}(\hat{H}-\langle\hat{H}\rangle_{\theta})|\varphi_{\theta}\rangle\langle\varphi_{\theta}|-|\varphi_{\theta}\rangle\langle\varphi_{\theta}|(\hat{H}^{\dagger}-\langle\hat{H}^{\dagger}\rangle_{\theta})\Big{]}$
(B20) $\displaystyle=$
$\displaystyle-2i\Big{[}\delta\hat{H}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|-|\varphi_{\theta}\rangle\langle\varphi_{\theta}|\delta\hat{H}^{\dagger}\Big{]}.$

Therefore, we obtain

$\displaystyle\ket{g}$ $\displaystyle=$ $\displaystyle\delta
L_{\theta}\ket{\varphi_{\theta}}$ (B21) $\displaystyle=$
$\displaystyle-2i\Big{[}\big{(}\delta\hat{H}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|-|\varphi_{\theta}\rangle\langle\varphi_{\theta}|\delta\hat{H}^{\dagger}\big{)}$
$\displaystyle\ \ \ \ \ \
-\big{(}\langle\delta\hat{H}\rangle_{\theta}-\langle\delta\hat{H}^{\dagger}\rangle_{\theta}\big{)}\Big{]}\ket{\varphi_{\theta}}$
$\displaystyle=$
$\displaystyle-2i\delta\hat{H}\ket{\varphi_{\theta}}=-2i\ket{f}.$

Obviously, it satisfies Eq. (B13), so SLD satisfies the condition for optimal
measurement. However, not only Eq. (B20) satisfies the condition, we can also
find other operators that satisfies the condition. We just need to add a term
on Eq. (B20), whose average value
$\bra{\varphi_{\theta}}\bullet\ket{\varphi_{\theta}}$ and the result acting on
$\ket{\varphi_{\theta}}$ are $0$, then the result of Eq. (B21) will not be
changed. For example,

$\displaystyle L^{\prime}_{\theta}$ $\displaystyle=$
$\displaystyle-2i\Big{[}\delta\hat{H}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|-|\varphi_{\theta}\rangle\langle\varphi_{\theta}|\delta\hat{H}^{\dagger}\Big{]}$
$\displaystyle+C\delta\hat{M}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|\delta\hat{M}^{\dagger},$

where $C$ is a real number and $\hat{M}$ is an arbitrary operator. Obviously,
$L^{\prime}_{\theta}$ also satisfies the condition for optimal measurement and
the equation
$(L^{\prime}_{\theta}\tilde{\rho}_{\theta}+\tilde{\rho}_{\theta}L^{\prime}_{\theta})/2=\partial_{\theta}\tilde{\rho}_{\theta}$.
Thus, we can find more optimal measurements with our condition.

## APPENDIX C: QFI of a specific $\mathcal{PT}$-symmetric non-Hermitian
Hamiltonian

Here, we provide the detailed calculations of the QFI for a specific
$\mathcal{PT}$-symmetric Hamiltonian based on Eq. (A15). For a given two level
$\mathcal{PT}$-symmetric Hamiltonian:

$\displaystyle\hat{H}_{s}=\begin{pmatrix}re^{i\omega}&s\\\
s&re^{-i\omega}\end{pmatrix}.$ (C1)

where $s>0$, $r$ and $\omega$ are real. We can see that the eigenvalues
$\lambda_{\pm}=\mu\pm\sqrt{\nu_{0}}=r\cos{\omega}\pm\sqrt{s^{2}-r^{2}\sin^{2}{\omega}}$
are still real when the $\mathcal{PT}$-symmetry is not broken, i.e.
$s^{2}>r^{2}\sin^{2}{\omega}$. And the normalized eigenstates are

$\displaystyle\ket{\lambda_{+}}=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\alpha/2}\\\
e^{-i\alpha/2}\end{pmatrix},\quad\ket{\lambda_{-}}=\frac{i}{\sqrt{2}}\begin{pmatrix}e^{-i\alpha/2}\\\
-e^{i\alpha/2}\end{pmatrix}.$

where $\sin{\alpha}=(r/s)\sin{\omega}$. As discussed above, it is necessary to
derive the normalized coefficient $K_{\theta}$. For an arbitrary initial state
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+me^{i\varphi}\ket{\lambda_{-}})$ written
in $\hat{H}_{s}$-present, we are able to work out the normalized coefficient
$K_{\theta}$ as follows:

$\displaystyle K_{\theta}^{(r)}$ $\displaystyle=$
$\displaystyle\bra{\psi_{0}}\hat{U}_{\theta}^{\dagger}\hat{U}_{\theta}\ket{\psi_{0}}$
(C2) $\displaystyle=$ $\displaystyle
N^{2}\\{e^{i(\lambda_{+}^{\ast}-\lambda_{+})\theta}+m^{2}e^{i(\lambda_{-}^{\ast}-\lambda_{-})\theta}+$
$\displaystyle
2m\textbf{Re}[e^{i[(\lambda_{+}^{\ast}-\lambda_{-})\theta+\varphi]}\lambda_{+}^{\ast}\lambda_{-}\langle\lambda_{+}|\lambda_{-}\rangle]\\}$
$\displaystyle=$ $\displaystyle N^{2}\\{1+m^{2}+2m\sin\alpha\cos\gamma\\},$

where $\gamma=2\sqrt{\nu_{0}}\theta+\varphi$. Let
$N_{\theta}^{(r)}=N^{2}/K_{\theta}^{(r)}$, then
$\langle\hat{H}_{s}^{\dagger}\hat{H}_{s}\rangle$ and
$\langle\hat{H}_{s}\rangle$ can be written as

$\displaystyle\langle\hat{H}_{s}^{\dagger}\hat{H}_{s}\rangle_{\theta}$
$\displaystyle=$ $\displaystyle
N_{\theta}^{(r)}(|\lambda_{+}|^{2}+m^{2}|\lambda_{-}|^{2}+$ $\displaystyle
2\textbf{Re}[m\lambda_{+}^{\ast}\lambda_{-}e^{i[(\lambda_{+}^{\ast}-\lambda_{-})\theta+\varphi]}\langle\lambda_{+}|\lambda_{-}\rangle])$
$\displaystyle=$ $\displaystyle
N_{\theta}^{(r)}[(r\cos\omega+\sqrt{\nu_{0}})^{2}+m^{2}(r\cos\omega-\sqrt{\nu}_{0})^{2}$
$\displaystyle+2m(r^{2}\cos^{2}\omega-\nu_{0})\sin\alpha\cos\gamma],$
$\displaystyle\langle\hat{H}_{s}^{\dagger}\rangle_{\theta}$ $\displaystyle=$
$\displaystyle
N_{\theta}^{(r)}\\{\lambda_{+}+m^{2}\lambda_{-}+m\sin\alpha(\lambda_{+}e^{i\gamma}+\lambda_{-}e^{-i\gamma})$
(C3) $\displaystyle=$ $\displaystyle
N_{\theta}^{(r)}[(r\cos\omega+\sqrt{\nu_{0}})+m^{2}(r\cos\omega-\sqrt{\nu}_{0})+$
$\displaystyle 2m\sin\alpha(r\cos\omega\cos\gamma+i\sqrt{\nu_{0}}\sin\gamma)]$

According to Eq. (A15), we have

$\displaystyle\mathcal{F}_{\theta}^{(r)}=\frac{16m^{2}(s^{2}-r^{2}\sin^{2}{\omega})^{2}}{[s(1+m^{2})+2mr\sin\omega\cos(2\sqrt{\nu_{0}}\theta+\varphi)]^{2}}.$

Figure 5: The evolution of QFI $\mathcal{F}_{\theta}^{(r)}$ for
$\mathcal{PT}$-symmetric Hamiltonian as a function of $\theta$. We set that
$s=1$, $r=0.4$, $\omega=\pi/2$, $m=1$ and $\varphi$ changes from $0$ to
$2\pi/3$, where $\varphi=$($2\pi/3,\pi/2,\pi/3,\pi/6,0$) (curves from left to
right).

In the case of $\mathcal{PT}$-symmetry broken, the eigenvalues turn to
$\varepsilon_{\pm}=\mu\pm\sqrt{\nu_{1}}=r\cos{\omega}\pm
i\sqrt{r^{2}\sin^{2}{\omega}-s^{2}}$, and the normalized eigenstates are

$\displaystyle\ket{\varepsilon_{\pm}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(r\sin\omega\pm\sqrt{\nu_{1}})^{2}+t^{2}}}\begin{pmatrix}i(r\sin\omega\pm\sqrt{\nu_{1}})\\\
t\end{pmatrix}.$

Let $\kappa=r\sin\omega/s$, for an arbitrary initial state
$\ket{\psi_{0}}=N(\ket{\varepsilon_{+}}+me^{i\varphi}\ket{\varepsilon_{-}})$,
similarly, we can obtain

$\displaystyle N_{\theta}^{(i)}$ $\displaystyle=$
$\displaystyle\frac{|N|^{2}}{K_{\theta}^{(i)}}=\frac{1}{e^{2\sqrt{\nu_{1}}\theta}+m^{2}e^{-2\sqrt{\nu_{1}}\theta}+2m\xi\cos\varphi},$
$\displaystyle\langle\hat{H}_{s}^{\dagger}\hat{H}_{s}\rangle_{\theta}$ (C7)
$\displaystyle=$ $\displaystyle
N_{\theta}^{(i)}\\{|\varepsilon_{+}|^{2}e^{i(\varepsilon_{+}^{\ast}-\varepsilon_{+})\theta}+m^{2}|\varepsilon_{-}|^{2}e^{i(\varepsilon_{-}^{\ast}-\varepsilon_{-})\theta}$
$\displaystyle+2\textbf{Re}(me^{i[(\varepsilon_{+}^{\ast}-\varepsilon_{+})\theta+\varphi]}\varepsilon_{+}^{\ast}\varepsilon_{-}\xi)\\}$
$\displaystyle=$ $\displaystyle
N_{\theta}^{(i)}\\{(r^{2}-s^{2})e^{2\sqrt{\nu_{1}}\theta}+m^{2}(r^{2}-s^{2})e^{-2\sqrt{\nu_{1}}\theta}$
$\displaystyle+2m\xi[(r^{2}\cos
2\omega+s^{2})\cos\varphi+2r\sqrt{\nu_{1}}\cos\omega\sin\varphi]\\},$
$\displaystyle\langle\hat{H}_{s}^{\dagger}\rangle_{\theta}$ $\displaystyle=$
$\displaystyle
N_{\theta}^{(i)}[\varepsilon_{+}^{\ast}e^{2\sqrt{\nu_{1}}\theta}+m^{2}\varepsilon_{-}^{\ast}e^{-2\sqrt{\nu_{1}}\theta}+2m\xi\textbf{Re}(\varepsilon_{+}^{*}e^{i\varphi})]$
$\displaystyle=$ $\displaystyle
N_{\theta}^{(i)}[(r\cos\omega-i\sqrt{\nu_{1}})e^{2\sqrt{\nu_{1}}\theta}$
$\displaystyle+m^{2}(r\cos\omega+i\sqrt{\nu_{1}})e^{-2\sqrt{\nu_{1}}\theta}$
$\displaystyle+2m\xi(r\cos\omega\cos\varphi+\sqrt{\nu_{1}}\sin\varphi)],$

where $\xi=\langle\varepsilon_{+}|\varepsilon_{-}\rangle=1/|\kappa|$.
Collecting these results, we arrive at

$\displaystyle\mathcal{F}_{\theta}^{(i)}=\frac{16m^{2}(r^{2}\sin^{2}{\omega}-s^{2})^{2}e^{4\sqrt{\nu_{1}}\theta}}{[A(e^{4\sqrt{\nu_{1}}\theta}+m^{2})+2mse^{2\sqrt{\nu_{1}}\theta}\cos\varphi]^{2}},$
(C8)

where $A=|r\sin\omega|$. And Fig. 5 shows that, for the given Hamiltonian, the
curve of QFI shifts along the $\theta-$axis with the change of $\varphi$. As
discussed in section V, we know that QFI of $\mathcal{PT}$-symmetric
Hamiltonian $\hat{H}_{s}$ can be much larger than that of Hermitian
Hamiltonian in some specific intervals of $\theta$, what if the actual
parameter $\theta_{0}$ is in the intervals that QFI of $\hat{H}_{s}$ is
smaller than that of Hermitian Hamiltonians? According to Eq. (APPENDIX C: QFI
of a specific $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian), the
relative phase $\varphi$ is only contained in
$\cos(2\sqrt{\nu_{0}}\theta+\varphi)$, hence $\varphi$ has nothing to do with
the the amplitude and frequency of $\mathcal{F}_{\theta}^{(r)}$. As shown in
Fig. 5, we can shift the curve of QFI to maximize it for a specific parameter
$\theta_{0}$ by adjusting the relative phase $\varphi$.

Then we discuss the channel QFI and why it does not reduce to zero as EP is
approached. In the two cases of $\mathcal{PT}-$symmetry broken and not, the
QFI with optimal initial states respectively are

$\displaystyle\mathcal{F}_{\theta}^{(r)}$ $\displaystyle=$
$\displaystyle\frac{4(s^{2}-r^{2}\sin^{2}{\omega})^{2}}{[s+r\sin\omega\cos(2\sqrt{\nu_{0}}\theta+\varphi)]^{2}},$
(C9) $\displaystyle\mathcal{F}_{\theta}^{(i)}$ $\displaystyle=$
$\displaystyle\frac{16m^{2}(r^{2}\sin^{2}{\omega}-s^{2})^{2}}{[A(e^{2\sqrt{\nu_{1}}\theta}+m^{2}e^{-2\sqrt{\nu_{1}}\theta})+2ms)]^{2}},$
(C10)

where $A=|r\sin\omega|$. For $\mathcal{F}_{\theta}^{(r)}$, if $r\sin\omega>0$,
it is maximum when $\cos(2\sqrt{\nu_{0}}\theta+\varphi)=-1$, then we have

$\displaystyle\mathcal{F}_{\theta,max}^{(r)}$ $\displaystyle=$
$\displaystyle\frac{4(s+r\sin\omega)^{2}(s-r\sin\omega)^{2}}{(s-r\sin\omega)^{2}}$
(C11) $\displaystyle=$ $\displaystyle 4(s+r\sin\omega)^{2}.$

The system reaches EP when $s-r\sin\omega=0$. However, $(s-r\sin\omega)^{2}$
in the $\mathcal{F}_{\theta,max}^{(r)}$ is omitted, so
$\mathcal{F}_{\theta,max}^{(r)}$ will not reduce to zero as EP is approached.
If $r\sin\omega<0$, $\mathcal{F}_{\theta}^{(r)}$ is maximum when
$\cos(2\sqrt{\nu_{0}}\theta+\varphi)=1$, we have

$\displaystyle\mathcal{F}_{\theta,max}^{(r)}$ $\displaystyle=$
$\displaystyle\frac{4(s+r\sin\omega)^{2}(s-r\sin\omega)^{2}}{(s+r\sin\omega)^{2}}$
(C12) $\displaystyle=$ $\displaystyle 4(s-r\sin\omega)^{2}.$

Similarly, the system reaches EP when $s+r\sin\omega=0$, but
$(s+r\sin\omega)^{2}$ in $\mathcal{F}_{\theta,max}^{(r)}$ is omitted. To sum
up, $\mathcal{F}_{\theta,max}^{(r)}$ can be written as

$\displaystyle\mathcal{F}_{\theta,max}^{(r)}=4(s+|r\sin\omega|)^{2},$ (C13)

it will not reduce to zero as EP is approached.

As for $\mathcal{F}_{\theta}^{(i)}$, it is maximum when
$\theta=\ln(-m)/2\sqrt{\nu_{1}}$, then we have

$\displaystyle\mathcal{F}_{\theta,max}^{(i)}$ $\displaystyle=$
$\displaystyle\frac{4(r^{2}\sin^{2}{\omega}-s^{2})^{2}}{(s-|r\sin\omega|)^{2}}.$
(C14) $\displaystyle=$
$\displaystyle\frac{4(r^{2}\sin^{2}{\omega}-s^{2})^{2}B^{2}}{(r^{2}\sin^{2}{\omega}-s^{2})^{2}}=4B^{2},$

where $B=s+|r\sin\omega|$. Similarly, the system reaches EP when
$r^{2}\sin^{2}{\omega}-s^{2}=0$, but $(r^{2}\sin^{2}{\omega}-s^{2})^{2}$ is
omitted in $\mathcal{F}_{\theta,max}^{(i)}$, so
$\mathcal{F}_{\theta,max}^{(i)}$ will not reduce to zero near EP.

## APPENDIX D: Optimal measurement and sensor of the specific
$\mathcal{PT}$-symmetric Hamiltonian

In the main text, we give an optimal measurement $|0\rangle\langle 0|$ for the
$\mathcal{PT}$-symmetric non-Hermitian Hamiltonian $\hat{H}_{s}$ in the case
of PT symmetry not broken, in this section, we are going to prove that the
measurement $|0\rangle\langle 0|$ satisfies the condition Eq. (B13). Here, we
denote $\hat{X}=|0\rangle\langle 0|$ and use $\kappa=r/s$, and remember that
we set $\omega=\pi/2$. As discussed in the main text, the measurement is
optimal only for the optimal initial state. In the case of
$\mathcal{PT}$-symmetry not broken, the optimal initial state is
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+e^{i\varphi}\ket{\lambda_{-}})$, then we
can obtain $\langle\hat{H_{s}}\rangle_{\theta}$ and
$\langle\hat{X}\rangle_{\theta}$ as follows:

$\displaystyle\langle\hat{H_{s}}\rangle_{\theta}$ $\displaystyle=$
$\displaystyle-i2N_{\theta}^{(r)}\nu_{0}[\kappa\sqrt{1-\kappa^{2}}\cos(2\sqrt{\nu_{0}}\theta+\varphi)+$
$\displaystyle\kappa^{2}\sin(2\sqrt{\nu_{0}}\theta+\varphi)],$
$\displaystyle\langle\hat{X}\rangle_{\theta}$ $\displaystyle=$ $\displaystyle
N_{\theta}^{(r)}\\{1+[(2\kappa^{2}-1)\cos(2\sqrt{\nu_{0}}\theta+\varphi)-$
(D1) $\displaystyle
2\kappa\sqrt{1-\kappa^{2}}\sin(2\sqrt{\nu_{0}}\theta+\varphi)]\\},$

According to Eq. (B8) and Eq. (B9), we have

$\displaystyle\ket{f}$ $\displaystyle=$
$\displaystyle(\hat{H_{s}}-\langle\hat{H_{s}}\rangle)\ket{\varphi_{\theta}}$
(D2) $\displaystyle=$
$\displaystyle\frac{4e^{i\frac{\varphi}{2}}N_{\theta}^{(r)}\nu_{0}}{s}\sqrt{\frac{N_{\theta}^{(r)}}{2}}\begin{pmatrix}\cos\gamma_{0}\\\
i\sin(\gamma_{0}+\beta)\end{pmatrix},$ $\displaystyle\ket{g}$ $\displaystyle=$
$\displaystyle(\hat{X}-\langle\hat{X}\rangle)\ket{\varphi_{\theta}}$
$\displaystyle=$ $\displaystyle
4e^{i\frac{\varphi}{2}}N_{\theta}^{(r)}\sqrt{\frac{N_{\theta}^{(r)}}{2}}\begin{pmatrix}i\cos^{2}\gamma_{0}\sin(\gamma_{0}+\beta)\\\
-\cos\gamma_{0}\sin^{2}(\gamma_{0}+\beta)\end{pmatrix},$

where $\gamma_{0}=\sqrt{\nu_{0}}\theta+\varphi/2$ and $\beta$ is determined by
$\kappa$ ($\sin\beta=\kappa,\cos\beta=-\sqrt{1-\kappa^{2}}$). Then we obtain

$\displaystyle\ket{f}=-\frac{i\nu_{0}}{s\cos\gamma_{0}\sin(\gamma_{0}+\beta)}\ket{g}.$
(D4)

Obviously, this result satisfies the condition $\ket{f}=iC\ket{g}$. Therefore,
$|0\rangle\langle 0|$ is indeed the optimal measurement only for the optimal
initial state.

Figure 6: Quantum sensor of $\mathcal{PT}$-symmetric non-Hermitian
Hamiltonian. (a) The expectation value $\langle\hat{X}\rangle$ as functions of
evolution parameter $\theta$, we change the value of $s$, so that the system
gradually approaches EP. We set $r=2,\omega=\pi/2$ and the initial state is
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+\ket{\lambda_{-}})$, the system reaches EP
when $s=2$, where orange dot-dashed line ($s=2.01$), black dashed line
($s=2.1$) and blue solid line ($s=3$). (b) The QFI as functions of evolution
parameter $\theta$, the parameter setting is the same as (a).

As discussed in the main text, the QFI $\mathcal{F}_{\theta}$ is always zero
at EP if the initial states can be expressed with eigenstates, so we can
speculate that the sensitivity of sensors reduces as EP is approached. We use
the optimal measurement $\hat{X}=|0\rangle\langle 0|$ as a sensor to verify
it, keep the initial state optimal, i.e.,
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+\ket{\lambda_{-}})$, the expectation of
$\hat{X}$ is

$\displaystyle\langle\hat{X}\rangle_{\theta}=\frac{1-\sin(2\theta\sqrt{s^{2}-r^{2}\sin^{2}\omega}-\alpha)}{2+2\sin\alpha\cos(2\theta\sqrt{s^{2}-r^{2}\sin^{2}\omega})}.$
(D5)

In Fig. 6(a), we show the expectation values $\langle\hat{X}\rangle_{\theta}$
and the QFI with different values of $s$. We can see that
$\langle\hat{X}\rangle$ is less sensitive to the changes in parameter
$\theta$, as the EP is approached, i.e., the sensitivity of sensor $\hat{X}$
reduces near EP. Note that in Fig. 6(a), there are still some intervals that
$\hat{X}$ is sensitive to the parameter $\theta$. As we discussed above, the
channel QFI $\mathcal{F}_{\theta,max}$ will not reduce to zero near EP, it
corresponds to the intervals that $\hat{X}$ is sensitive to parameter
$\theta$. Comparing Fig. 6(a) with Fig. 6(b), we can see that the intervals
with high sensor sensitivity corresponds to the intervals with large QFI. The
performance of sensor $\hat{X}$ is consistent with our analysis of QFI.

Figure 7: The evolution of ratio $S_{\theta}$ as a function of $\theta$. We
set that $s=1$, $\kappa$ changes from $0.2$ to $0.6$ with an interval of
$0.2$, and the initial state
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+\ket{\lambda_{-}})$ remains unchanged. (a)
The ratio factor $S_{\theta}^{(0)}$ of systems with different modulus of
energy, where $n=$ ($0.2,0.3,0.4,0.5,0.6$) (from bottom to top). (b) The ratio
factor $S_{\theta}^{(1)}$ of systems with different modulus of energy, where
$n=$ ($0.2,0.3,0.4,0.5,0.6$) (curves from top to bottom).

## APPENDIX E: Comparison of Hermitian and non-Hermitian Hamiltonian QFI

Knowing the QFI of $\hat{H}_{s}$, we compare $\mathcal{F}_{\theta}^{(r)}$ with
the QFI of two Hermitian Hamiltonians, for simplicity, we set $\omega=\pi/2$.
The Hermitian Hamiltonians are

$\displaystyle\hat{H}_{0}=\begin{pmatrix}\sqrt{s^{2}-r^{2}}&0\\\
0&-\sqrt{s^{2}-r^{2}}\end{pmatrix},\hat{H}_{1}=\begin{pmatrix}s&-ir\\\
ir&s\end{pmatrix}.$

Note that $\hat{H}_{0}$ has the same eigenvalues with $\mathcal{PT}$-symmetric
Hamiltonian $\hat{H}_{s}$, and
$\hat{H}_{1}^{\dagger}\hat{H}_{1}=\hat{H}_{s}^{\dagger}\hat{H}_{s}$. That is
why we select these two Hermitian Hamiltonians. It is not difficult to
calculate the corresponding optimal QFI, we have
$\mathcal{F}_{\theta}^{(0)}=4\lambda_{0}^{2}$ and
$\mathcal{F}_{\theta}^{(1)}=4r^{2}$. We find that the QFI of Hamiltonian
$\hat{H}_{s}$ can be much larger than that of two Hermitian Hamiltonians. To
illustrate this, we define the ratios $S_{\theta}$ as functions as follows:

$\displaystyle S_{\theta}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{F}_{\theta}^{(r)}}{\mathcal{F}_{\theta}^{(0)}}=\frac{4m^{2}(1-\kappa^{2})}{[1+m^{2}+2m\kappa\sin(2\sqrt{\nu_{0}}\theta+\varphi+\alpha)]^{2}},$
$\displaystyle S_{\theta}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{F}_{\theta}^{(r)}}{\mathcal{F}_{\theta}^{(1)}}=\frac{4m^{2}(1-\kappa^{2})^{2}}{\kappa^{2}[1+m^{2}+2m\kappa\sin(2\sqrt{\nu_{0}}\theta+\varphi+\alpha)]^{2}}.$

Let us consider an example of the comparison, we set the initial state as
$\ket{\psi_{0}}=N(\ket{\lambda_{+}}+\ket{\lambda_{-}})$ which is optimal, and
change the modulus of energy $\lambda_{0}$. If there is $S_{\theta}>1$, our
conjecture is verified. We can see that in Fig. 7(a) and (b), the ratios
$S_{\theta}^{(0)}$ and $S_{\theta}^{(1)}$ change with the value of $n$, and
the curves above the grey dotted line indicate that the QFI for non-Hermitian
system can be larger than that of Hermitian systems in some intervals, among
them, the maximum value of $S_{\theta}^{(0)}$ can reach 4, while the maximum
value of $S_{\theta}^{(1)}$ can even reach 36! As mentioned in section III, as
we change the relative phase $\varphi$ in initial state $\ket{\psi_{0}}$, the
curves shift along the $\theta-$axis, so that we can guarantee that
$\mathcal{F}_{\theta}^{(r)}$ can be larger than $\mathcal{F}_{\theta}^{(0)}$
or $\mathcal{F}_{\theta}^{(1)}$ in arbitrary intervals of $\theta$.

## APPENDIX F: QFI of a two coupled bosonic modes system

In most cases, the dynamic of an open system is described with master
equation, however, if the quantum jump terms can be neglect, the system can be
described with an effective non-Hermitian Hamiltonian $H_{eff}$ qtj . Our
results is still feasible for such a system.

For example, let us consider a two coupled bosonic modes system

$\displaystyle\hat{H}=\omega(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b})+g(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a}),$
(F1)

and the dynamic is

$\displaystyle\frac{\partial\hat{\rho}(t)}{\partial
t}=-i[\hat{H},\hat{\rho}(t)]+(\frac{\gamma_{a}}{2}\mathcal{D}[\hat{a}]+\frac{\gamma_{b}}{2}\mathcal{D}[\hat{b}])\hat{\rho}(t),$

where the dissipator
$\mathcal{D}[\hat{a}]\hat{\rho}(t)=\hat{a}\hat{\rho}(t)\hat{a}^{\dagger}-\\{\hat{a}^{\dagger}\hat{a},\hat{\rho}(t)\\}/2$.
In the semiclassical limit, this system can be described with a non-Hermitian
Hamiltonian as follow qtj

$\displaystyle\hat{H}_{eff}=(\omega-\frac{i\gamma_{a}}{2})\hat{a}^{\dagger}\hat{a}+(\omega-\frac{i\gamma_{b}}{2})\hat{b}^{\dagger}\hat{b}+g(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a}).$

In the case of one excitation, the eigenvalues of the system are
$\lambda_{\pm}=\omega-i\overline{\gamma}/2\pm\xi$, where
$\overline{\gamma}=(\gamma_{a}+\gamma_{b})/2$, $\xi^{2}=g^{2}-\gamma^{2}/4$
and $\gamma=(\gamma_{a}-\gamma_{b})$. Then the normalized eigenstates can be
expressed with Fock state as follow

$\displaystyle|\lambda_{\pm}\rangle=\frac{1}{\sqrt{2}g}[(-i\frac{\gamma}{2}\pm\xi)|0,1\rangle+g|1,0\rangle].$
(F4)

The dynamic of this non-Hermitian Hamiltonian is
$d\hat{\rho}_{\theta}/d\theta=-i(\hat{H}_{eff}\hat{\rho}_{\theta}-\hat{\rho}_{\theta}\hat{H}_{eff}^{\dagger})$,
then the evolution operator is $\hat{U}_{\theta}=e^{-i\hat{H}\theta}$. Here,
we assume that $\xi^{2}=g^{2}-\gamma^{2}/4\geq 0$, for a given normalized
initial state
$|\psi_{0}\rangle=(|\lambda_{+}\rangle-|\lambda_{-}\rangle)g/(\sqrt{2}\xi)=|0,1\rangle$,
the QFI is

$\displaystyle\mathcal{F}_{\theta}$ $\displaystyle=$ $\displaystyle
4(\langle\varphi_{\theta}|H^{\dagger}H|\varphi_{\theta}\rangle-\langle\varphi_{\theta}|H^{\dagger}|\varphi_{\theta}\rangle\langle\varphi_{\theta}|H|\varphi_{\theta}\rangle)$
(F5) $\displaystyle=$
$\displaystyle\frac{64g^{2}\xi^{4}}{[-4g^{2}+\gamma^{2}\cos(2\xi\theta)+2\gamma\xi\sin(2\xi\theta)]^{2}}.$

Therefore, our result can also be used for open systems which can be described
with an effective non-Hermitian Hamiltonian in the semiclassical limit.

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