Hermitian and non-Hermitian Hamiltonian are the two types of quantum systems. It is well known that the dynamics of systems governed by Hermitian Hamiltonian are referred to as unitary. In this case, the Hermitian Hamiltonian is characterized by real energy spectrum. However, Non-Hermitian Hamiltonian system undergoes non-unitary evolution which corresponds to an open quantum system influenced by an external environment. In such systems, it provides complex eigenvalues and non-orthogonal eigenvectors. Due to the rapid development of non-Hermitian optical and micro-cavity experiments^[1], a class of non-Hermitian Hamiltonian with parity-time(PT) symmetry exhibiting real energy eigenvalues have attracted numerous attentions.

It has been demonstrated that non-Hermitian systems in the parameter space usually experience the quantum criticality at an exceptional point (EP). Arising from the coalescence of both eigenvalues and eigenvectors, non-Hermitian systems with PT symmetry exhibit quantum phase transition from unbroken phase regime to broken one. In recent years, some new features at exceptional points can be observed in non-Hermitian system, such as single-mode lasing^[2], enhancing optical response behaviors^[3], PT-symmetry-breaking chaos^[4], flexible switching for light transmission^[5], loss-induced transparency^[6], and EP-enhanced sensing^[7-8]. Therefore, a fundamental issue is how to effectively detect and characterize critical phenomena in non-Hermitian systems^[9].

In some works, various kinds of witnesses have been put forward to capture the features of quantum phase transitions, including Hilbert-Schmidt speed^[10], quantum coherence^[11] and entropy^[12]. Although these quantities are sensitive to quantum criticality, they are theoretical evidences which are indirectly observable. To directly measure critical phenomena, we expect to apply metrological methods from the viewpoint of parameter estimation. This motivation stimulates us to explore the relationship between quantum metrology and quantum phase transition in non-Hermitian systems. It is well known that quantum Fisher information (QFI) is usually exploited to evaluate the precision of parameter estimation^[13]. Consequently, an interesting question arises: how to determine the PT phase transitions from the perspective of quantum parameter estimation. In the present work, we provide the witness based on quantum Fisher information to accurately characterize signatures of quantum critical transitions in non-Hermitian ion trap systems with PT symmetry.

This paper is organized as follows. In Sec. II, a normalized density matrix in the form of state vector is used to define quantum Fisher information in a non-Hermitian PT-symmetric system. In Sec. III, we provide a general formulation to characterize the non-unitary dynamics of any two-level non-Hermitian system. By using the general approach, we investigate the dynamics of quantum Fisher information in single ion trapped system. The physical relationship between quantum parameter estimation and quantum phase transition is studied in Sec. IV. The exceptional criticality is also verified by quantum entropy and quantum coherence. Finally, a conclusion is given in Sec VI.

To obtain the QFI for non-Hermitian system, let us start by reviewing some fundamental features of the QFI definition. For a quantum state $ {\rho _\theta } $ with an unknown parameter $ \theta $, the QFI $ {F_\theta } $ provides the ultimate precision limit according to the estimation theory. It is known as the quantum Cramér-Rao bound, $ V(\theta ) \geqslant 1/(n{F_\theta }) $ where $ n $ is the number of measurements. The large value QFI corresponds to the high precision of parameter estimation. The general expression is written as $ {F_\theta } = Tr[L_\theta ^2{\rho _\theta }] $ where the symmetric logarithmic derivative operator $ {L_\theta } $ satisfies $ ({L_\theta }{\rho _\theta } + {\rho _\theta }{L_\theta })/2 = {\partial _\theta }{\rho _\theta } $ and arbitrary states ${\rho _\theta }$ can be diagonalized^[14-15]. Note that a Hermitian system governs an unitary evolution which obeys the conservation of the trace of the density matrix and the eigenvectors are orthogonal and normalized. However, the dynamics of a non-Hermitian quantum system is no more unitary and the estimated state is no longer normalized. According to the derivation^[16], we need to normalize the evolved state as $ {\tilde \rho _\theta } = {\rho _\theta }/Tr({\rho _\theta }) $ where $ Tr({\rho _\theta }) $ is the trace of the estimated density matrix. In the condition of PT symmetry, the density matrix of a $ d - $level system is Hermitian which satisfies $\rho _\theta ^{{\dagger }} = {\rho _\theta }$, and then can be expressed as a linear superposition of a complete set of Hermitian operators. In the case of $ d = 2 $, ${\rho _\theta }(t) = \dfrac{1}{2} \displaystyle\sum _{{{j = 0}}}^{\text{3}} {\lambda _j}\left( t \right){\hat E_j},$ where the coefficients ${\lambda _j}(t) = Tr\left[ {{\rho _\theta }\left( t \right){{\hat E}_j}} \right]$ are real. Here, {${\hat E_j}$}={${\sigma _{\text{0}}}{\text{,}}{\sigma _{\text{1}}}{\text{,}}{\sigma _{\text{2}}}{\text{,}}{\sigma _{\text{3}}}$} is a complete set of Pauli operators and ${Tr({\rho _\theta })} = {\lambda _0}(t)$ is the normalized parameter. If the normalized matrix is ${\tilde \rho _\theta } = \displaystyle\sum\limits_j {{c_j}} \left| {{\psi _j}} \right\rangle \left\langle {{\psi _j}} \right|$ , the QFI $ {F_\theta } $ is written as

$ {F_\theta } = 2\sum\limits_{j,k} {\frac{{{{\left| {\left\langle {{\psi _j}} \right|{\partial _\theta }{{\tilde \rho }_\theta }\left| {{\psi _k}} \right\rangle } \right|}^2}}}{{{c_j} + {c_k}}}}\quad, $ (1)

where the relation of eigenvalues $ {c_j} + {c_k} \ne 0 $ is required and ${c_{1,2}} = \dfrac{{{\lambda _0}\left( t \right) \pm \displaystyle\sum _{j = 1}^3 \sqrt {\lambda _j^2\left( t \right)} }}{{2{\lambda _0}\left( t \right)}}$, $\left| {{\psi _{1,2}}} \right\rangle = \left( {\begin{array}{*{20}{c}} {{\lambda _1}\left( t \right) - i{\lambda _2}\left( t \right)} \\ { - {\lambda _3}\left( t \right) \pm \displaystyle\sum_{j = 1}^3 \sqrt {\lambda _j^2\left( t \right)} } \end{array}} \right)$ denote the eigenvalues and eigenvectors of the estimated state respectively.

To demonstrate the dynamical behavior of QFI in a non-Hermitian system, we consider a general two-level non-Hermitian system whose Hamiltonian is written as

$ \hat H = {\hat H_ + } + {\hat H_ - }\quad, $ (2)

where a Hermitian part ${\hat H_ + } = {\hat H^\dagger }_ + = \displaystyle\sum {\alpha _k}{\hat E_k}$, an anti-Hermitian one ${\hat H_ - } = - {\hat H^\dagger }_ - = i\displaystyle\sum {\beta _l}{\hat E_l}$ and the coefficients $ \left\{ {{\alpha _k},{\beta _l}} \right\} $ are real. The state evolution of the non-Hermitian system is governed by the time-dependent Schrodinger equation^[17-18] which reads as,

$ \begin{split} \frac{{\partial \rho \left( t \right)}}{{\partial t}} =& - \frac{i}{\hbar }\left( {\hat H\rho \left( t \right) - \rho \left( t \right){{\hat H}^\dagger }} \right) =\\ &- \frac{i}{\hbar }\left[ {{{\hat H}_ + },\rho \left( t \right)} \right] - \frac{i}{\hbar }\left\{ {{{\hat H}_ - },\rho \left( t \right)} \right\} \quad, \end{split} $ (3)

here, the square bracket $ \left[ { \cdot , \cdot } \right] $ is the commutator, the curly bracket denotes the anti-commutator $ \left\{ { \cdot , \cdot } \right\} $ and the density matrix $\rho \left( t \right)$describes the state of the non-Hermitian system. By using the vector formalism $ \vec \lambda (t) $ of the density matrix, the dynamics of the non-Hermitian system can be expressed as,

$ \frac{\partial }{{\partial t}}\vec \lambda (t) = {{\hat {\boldsymbol{\zeta }}}} \cdot \vec \lambda (t) \quad,$ (4)

where the dynamical mapping matrix is $\hat {\boldsymbol{\zeta }}= \dfrac{{\text{2}}}{\hbar }\left( {\begin{array}{*{20}{c}} {{\beta _{\text{0}}}}&{{\beta _{\text{1}}}}&{{\beta _{\text{2}}}}&{{\beta _{\text{3}}}} \\ {{\beta _{\text{1}}}}&{{\beta _{\text{0}}}}&{{\alpha _{\text{3}}}}&{{{ - }}{\alpha _{\text{2}}}} \\ {{\beta _{\text{2}}}}&{{\alpha _{\text{3}}}}&{{\beta _{\text{0}}}}&{{{ - }}{\alpha _{\text{1}}}} \\ {{\beta _{\text{3}}}}&{{{ - }}{\alpha _{\text{2}}}}&{{\alpha _{\text{1}}}}&{{\beta _{\text{0}}}} \end{array}} \right)$ and the evolved vector $ \vec \lambda (t) = {({\lambda _0},{\lambda _1},{\lambda _2},{\lambda _3})^{\mathrm{T}}} $ describing the evolved state at any time. In general, the state vector $\vec \lambda \left( t \right)$ is given by $\vec \lambda \left( t \right){\text{ = }}{e^{\hat \zeta t}} \cdot \vec \lambda \left( {\text{0}} \right)$ and $\vec \lambda \left( 0 \right)$ represents the vector of an initial state. With respect to the diagonal form of $\hat \zeta = \displaystyle\sum\limits_{i = 1}^4 {{h_i}\left| {{\nu _i}} \right\rangle \left\langle {{\nu _i}} \right|} $ , the exponential function is given by ${e^{\hat \zeta t}} = \hat V\hat R{\hat V^{ - 1}}$ and the i-th column of the transformation matrix $\hat V$ is the corresponding eigenvector $ \left| {{\nu _i}} \right\rangle $. The diagonal matrix $ \hat R ={\mathrm{diag}}({e^{{h_1}t}},{e^{{h_2}t}},{e^{{h_2}t}}, {e^{{h_4}t}}) $ is determined by the eigenvalues of the mapping matrix.

To study quantum dynamics of a non-Hermitian system, we take into account a single-ion system where the PT-symmetric Hamiltonian with balanced gain and loss has been realized in the experiment^[19]. The experimental setup involves a trapped 40Ca⁺ ion in a magnetic field. Initially prepared in the ground state $\left| g \right\rangle = \left| {{}^2{S_{ - 1/2}}} \right\rangle $, the system is excited to an upper state $\left| e \right\rangle = \left| {{}^2{D_{5/2}}} \right\rangle $ using a laser with a wavelength of 729 nm. Another laser at 854 nm induces a controllable loss in $\left| e \right\rangle $ which is coupled to a short-lived level $\left| {{}^2{P_{3/2}}} \right\rangle $ and quickly decays to the state $\left| d \right\rangle = \left| {{}^2{S_{1/2}}} \right\rangle $. By means of laser driving, the coherent energy transition between $\left| e \right\rangle $ and $\left| g \right\rangle $ can be performed and the excited state undergoes the tunable loss, as shown in Figure 1. From the viewpoint of open quantum systems, the states of $\left| e \right\rangle $ and $\left| g \right\rangle $ experience the interaction with the dissipation environment $\left| d \right\rangle $ at a loss rate.

In this case, the effective two-level system is described by the non-Hermitian Hamiltonian,

$ {{{\hat H}_{{\text{eff}}}} = \frac{\varOmega }{{\text{2}}}{\sigma _1} - i\frac{\gamma }{{\text{2}}}{\sigma _3} - i\frac{\gamma }{{\text{2}}}} I \quad,$ (5)

where $\gamma $ is the gain-loss rate and $\varOmega $ is the coherent coupling. Correspondingly, $ \vec{\alpha }{ =}{\left({0,}\dfrac{\varOmega }{{2}}{,0,0}\right)}^{{\mathrm{T}}}\text{, } \vec{\beta }= {\left({-}\dfrac{\gamma }{{2}}{,0,0,-}\dfrac{\gamma }{\text{2}}\right)}^{{\mathrm{T}}} $ can characterize Hermitian part and anti-Hermitian one of the non-Hermitian Hamiltonian in the trapped ion system. Additionally, some experimental setups^[20] of non-Hermitian system are realized in the field of optics and photonics.

Quantum phase transition in non-Hermitian systems is traditionally characterized by the energy spectrum of the Hamiltonian. Like the non-Hermitian Hamiltonian of Eq. (6), the PT symmetric part ${\hat H_{{\mathrm{PT}}}} = \dfrac{\varOmega }{2}{\sigma _1} - i\dfrac{\gamma }{2}{\sigma _3}$ of the Hamiltonian has two eigenvalues of ${E_{1,2}} = \pm \dfrac{1}{2}\sqrt {{\varOmega ^2} - {\gamma ^2}} $ where the region of $\dfrac{\gamma }{\varOmega } < 1$ corresponds to the unbroken phase of PT symmetry and $\dfrac{\gamma }{\varOmega } > 1$ represents the PT symmetry broken phase. The criticality occurs at an EP of $ \gamma = \varOmega $ due to the interplay between the gain-loss rate and coherent coupling.

Recently, some feasible approaches in quantum information theory can be used to describe the criticality at the exceptional points of non-Hermitian systems, such as quantum entanglement, quantum correlation and others. It is interesting to study the physical relation between quantum criticality and quantum parameter estimation in non-Hermitian systems. In the experimental condition, we can obtain the dynamics of the non-Hermitian single ion system by using Eq. (5). And the eigenvalues of ${{\hat \zeta }}$ are $ {h}_{1}=-\gamma $, ${h_2} = - \gamma $, ${h_{3,4}} = - \gamma \pm K$ and $K = \sqrt {{\gamma ^{\text{2}}}{{ - }}{\varOmega ^{\text{2}}}} $. The eigenvectors are given as $\left| {{v_1}} \right\rangle = \left( {\begin{array}{*{20}{c}} {\text{0}} \\ {\text{1}} \\ {\text{0}} \\ {\text{0}} \end{array}} \right)$, $\left| {{v_2}} \right\rangle = \left( {\begin{array}{*{20}{c}} {\text{1}} \\ {\text{0}} \\ {\lambda /\varOmega } \\ {\text{0}} \end{array}} \right)$, $\left| {{v_{3,4}}} \right\rangle = \left( {\begin{array}{*{20}{c}} {\text{1}} \\ {\text{0}} \\ {\varOmega /\gamma } \\ { \mp K/\gamma } \end{array}} \right)$ respectively. The initial state as a function of estimated parameters can be expressed as,

$ \rho \left( {\text{0}} \right) = \left( {\begin{array}{*{20}{c}} {{\text{co}}{{\text{s}}^{\text{2}}}\left( {\dfrac{\theta }{{\text{2}}}} \right)}&{{\text{sin}}\dfrac{\theta }{{\text{2}}}{\text{cos}}\dfrac{\theta }{{\text{2}}}{{\text{e}}^{{{ - }}i\phi }}} \\ {{\text{sin}}\dfrac{\theta }{{\text{2}}}{\text{cos}}\dfrac{\theta }{{\text{2}}}{{\text{e}}^{i\phi }}}&{{\text{si}}{{\text{n}}^{\text{2}}}\left( {\dfrac{\theta }{{\text{2}}}} \right)} \end{array}} \right)\quad, $ (6)

where $ \rho (0) = \left| {\phi (0)} \right\rangle \left\langle {\phi (0)} \right|,\left| {\phi (0)} \right\rangle = \cos \dfrac{\theta }{2}\left| e \right\rangle +\sin\dfrac{\theta }{2} {e^{i\varphi }}\left| g \right\rangle $. The estimated parameters of $ \theta ,\varphi $ are related to the directly measured density matrix elements. The equivalent form of the initial state vector is written as $ \vec \lambda (0) = ( 1,\sin \theta \cos \varphi ,\sin \theta \sin \varphi , \cos \theta )^{\mathrm{T}} $. At any time, the density matrix $ \rho (t) $ can be described by the state vector $ \vec \lambda (t) $ in the form of

$ \begin{split} & {{\lambda _0}\left( t \right) = {e^{ - \gamma t}}\left[ {\frac{{{\gamma ^2}}}{{{K^2}}}\cosh \left( {Kt} \right) - \frac{{{\varOmega ^2}}}{{{K^2}}} + \left( {\frac{{\gamma \varOmega }}{{{K^2}}}\cosh \left( {Kt} \right) + \frac{{\gamma \varOmega }}{{{K^2}}}} \right)\sin \theta \sin \phi - \frac{\gamma }{K}\sinh \left( {Kt} \right)\cos \theta } \right],} \\ & {{\lambda _1}\left( t \right) = {e^{ - \gamma t}}\sin \theta \cos \phi ,} \\ & {{\lambda _2}\left( t \right) = {e^{ - \gamma t}}\left[ {\frac{{\gamma \varOmega }}{{{K^2}}}\cosh \left( {Kt} \right) - \frac{{\gamma \varOmega }}{{{K^2}}} + \left( {\frac{{{\varOmega ^2}}}{{{K^2}}}\cosh \left( {Kt} \right) + \frac{{{\gamma ^2}}}{{{K^2}}}} \right)\sin \theta \sin \phi - \frac{\varOmega }{K}\sinh \left( {Kt} \right)\cos \theta } \right],} \\ & {{\lambda _3}\left( t \right) = {e^{ - \gamma t}}\left[ { - \frac{\gamma }{K}\sinh \left( {Kt} \right) - \frac{\varOmega }{K}\sinh \left( {Kt} \right)\sin \theta \sin \phi + \cosh \left( {Kt} \right)\cos \theta } \right]\quad.} \end{split} $ (7)

Without loss of generality, we consider the estimation precision of the phase parameter $ \phi = {\mathrm{arctan}}\left( {\dfrac{{{\lambda _2}}}{{{\lambda _1}}}} \right) $ which is bounded by the QFI,

$ \begin{array}{*{20}{c}} {{F_\phi } = {{\left| {{\partial _\phi }\vec B} \right|}^2} + } \end{array}{(\vec B \cdot {\partial _\phi }\vec B)^2}/(1 - |\vec B{|^2})\quad, $ (8)

here the vector is given by $ \vec B = \dfrac{1}{{{\lambda _0}}}{\left( {{\lambda _1},{\lambda _2},{\lambda _3}} \right)^{\mathrm{T}}} $ which is dependent on the normalized density matrix $ {\tilde \rho _\phi } = {\rho _\phi }/Tr({\rho _\phi }) $ with the estimated phase parameter.

To demonstrate the connection between QFI and quantum phase transition at EP, we explore the dynamics of QFI as a function of the ratio $\gamma /\varOmega $, which is shown in Fig. 2(a) (color online). It is seen that the values of QFI approach to the maximum near the EP of $\gamma = \varOmega $ at the early stage of the evolution. This result proves that the non-classical effects are amplified in the vicinity of the EP. In the unbroken region of PT symmetry, the oscillatory behavior of QFI can occur in Fig. 2(b). In fact, a non-Hermitian system can be referred to as one open system. The discontinuous revival behavior can arise from the non-Markovian dynamics which is characterized as information retrieval from the environment to the system. However, the QFI undergoes the monotonically decreasing evolution in the broken phase as shown in Fig. 2(c). The unidirectional information flow from the system to the environment leads to the monotonic decaying of QFI. Therefore, the EP masks the oscillation and monotonic declination of quantum Fisher information. The two different ways of the evolution of QFI can be regarded as a signature of quantum phase transition. Moreover, the more accurate estimation can be obtained in the unbroken phase of PT symmetry.

To furthermore study the effects of the balanced gain and loss on estimation precisions, we also illustrate the short-time behavior of QFI with respect to all possible values of the phase parameter. From Fig. 3 (color online), the contour plot of QFI as a function of $ \dfrac{\gamma }{\varOmega } $ and $ \varphi $ is shown in the condition of $ \varOmega = 2{\text{π}} \times 32 $ kHz, $ \theta = {\text{π}} /2 $, t = 19 μs. We can observe the values of QFI arrive at some peaks in the vicinity of the EP of $ \dfrac{\gamma }{\varOmega } = 1 $ . It is demonstrated that for some certain values of $ \varphi = n{\text{π}} (n = 0,1,2\cdots) $, the maximal values of the QFI are obtained at the exceptional point. This result suggests that near the EP, there is an enhanced availability of quantum estimation precision. The QFI is sensitive to quantum criticality at the EP.

It is also demonstrated that the non-Hermitian QFI about parameter estimation can be regarded as one feasible witness to determine the critical points of the system.

In the study of the dynamics of quantum systems, people often focus on the evolution of some facets of quantum information including quantum nonlocality. We expect to find more evidences to characterize quantum phase transition in a non-Hermitian system, from the viewpoint of quantum information. In the following, the dynamical behavior of quantum entropy and quantum coherence are examined to demonstrate the effects of quantum information facets on quantum criticality at the EP.

It is known that quantum entropy can reveal the flow and distribution of quantum information within the system. The characteristics of the von Neumann entropy can be obtained from the dynamics of the non-Hermitian system. For the non-Hermitian system, quantum entropy is defined by the normalized density matrix^[21,22]:

$ \begin{array}{*{20}{c}} {S\left( {\tilde \rho } \right) = - Tr\left[ {\tilde \rho \ln \tilde \rho } \right]} \quad. \end{array} $ (9)

When the single ion system is considered, the normalized density matrix is written as

$ \begin{array}{*{20}{c}} {\tilde \rho \left( t \right) = \left( {\begin{array}{*{20}{c}} {\dfrac{1}{2} + \dfrac{{{\lambda _3}}}{{2{\lambda _0}}}}&{\dfrac{{{\lambda _1} - i{\lambda _2}}}{{2{\lambda _0}}}} \\ {\dfrac{{{\lambda _1}{\text{ + }}i{\lambda _2}}}{{2{\lambda _0}}}}&{\dfrac{1}{2} - \dfrac{{{\lambda _3}}}{{2{\lambda _0}}}} \end{array}} \right)} \quad. \end{array} $ (10)

Here the initial state $ \rho (0) $ is chosen as the form of Eq. (6). The dynamical behavior of quantum entropy can be plotted in Fig. 4(a) (color online) when the parameters are $ \varOmega = 2{\text{π}} \times 32 $ kHz, $ \theta = {\text{π}} /2,\varphi = 0 $. It is shown that in the region of $ \dfrac{\gamma}{\varOmega} < 1 $, quantum entropy exhibits periodic oscillations with time. In contrast, for the PT symmetry broken region of $ \dfrac{\gamma}{\varOmega} > 1 $, the entropy gradually decreases and approaches to a stable value. From Fig. 4(b), we can observe that the time derivative $ dS/dt $ suddenly changes near the EP. The time is set as t = 66.8 μs.

The Fig. 4(b) shows the first derivative of entropy with respect to the ratio of $\dfrac{\gamma}{\varOmega} $ . From the figure, we can clearly distinguish three different regions. In the PTS region, the entropy has a maximum value implies the presence of substantial quantum resources. If the value of $ \dfrac{\gamma}{\varOmega} $ approaches to the EP, the time derivative of entropy suddenly changes to zero from negative values. The result also demonstrates that the maximum of quantum entropy can be accessed in the vicinity of the EP. It is proven that the dynamics of quantum entropy can be used to describe the critical phenomena of the non-Hermitian system.

Meanwhile, quantum coherence emerges when quantum states exist in the form of the superposition. It serves as one kind of fundamental resources in quantum information technology.

According to the definition of $ {l_1} $-norm^[23-24], quantum coherence can be calculated as $ C\left( {\tilde \rho } \right) = \displaystyle\sum _{i \ne j}^{} \left| {{{\tilde \rho }_{ij}}} \right| $ and determined by the off-diagonal elements of the normalized density matrix. The expression of quantum coherence for the non-Hermitian ion system can be given by

$ {{C_{l1}} = \frac{{\sqrt {\lambda _1^2{{ + }}\lambda _2^2} }}{{{\text{|}}{\lambda _0}{\text{|}}}}} \quad. $ (11)

The dynamical behavior of quantum coherence as a function of $ \dfrac{\gamma}{\varOmega} $ can be shown in Fig. 5 (color online) when the parameters are $ \varOmega = 2{\text{π}} \times 32 $ kHz, $ \theta = {\text{π}} /2,\phi = 0 $.

It is seen that the high values of coherence occur in an oscillatory manner in the PT symmetry unbroken region. With the increase of the loss $ \gamma $, quantum coherence decreases monotonically in the PT symmetry broken region. The EP marks the two different kinds of behavior from the viewpoint of quantum coherence.

Utilizing quantum metrology techniques, we provide a witness of quantum Fisher information to characterize the phase transition of a non-Hermitian system. The state vector of the normalized density matrix is used to express the dynamics of the non-Hermitian QFI. The single trapped ion system can be applied to our scheme. The two different types of dynamical behaviors are observed near the exceptional point. In the PT symmetry unbroken region, the oscillatory values of the QFI keep large, which represent the high precision of quantum parameter estimation. On the other hand, the monotonical declination of the QFI is demonstrated in the PT symmetry broken region. With the increase of the loss, the QFI gradually decreases towards a steady value which is dependent on the equilibrium state. The sudden change in the evolution of QFI at EP can be regarded as a signature of quantum criticality of the non-Hermitian system. In addition, we have considered the dynamical behavior of quantum entropy and coherence. From the viewpoint of quantum information, the discontinuous behavior can also be demonstrated at the EP. Much more quantum resources like quantum entropy and coherence can be obtained near the EP, which is beneficial for quantum information processing. Our work not only may provide a new way to enhancing parameter estimation precision but also discover the physical relation between quantum information and quantum criticality in a non-Hermitian system.