The generalized CV conjecture of Krylov complexity

Ke-Hong Zhai [email protected] Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, China Lei-Hua Liu [email protected] Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, China Hai-Qing Zhang [email protected] Center for Gravitational Physics, Department of Space Science, Beihang University, Beijing 100191, China Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China

Abstract

We extend the “complexity=volume” (CV) conjecture in the wormhole to the quantum states in the framework of information geometry. In particular, we conjecture that Krylov complexity equals the volume of the Fubini-Study metric in the information geometry. In order to test our conjecture, we study the general Hermitian two-mode Hamiltonian according to the Weyl algebra both in the closed and open systems. By employing the displacement operator, we find that the wave function for a closed system corresponds to the well-known two-mode squeezed state. For an open system, we can create a wave function known as the open two-mode squeezed state by using the second kind of Meixner polynomials. Remarkably, in both cases, the resulting volume of the corresponding Fubini-Study metric provides strong evidence for the generalized CV conjecture.

I Introduction

Recently, complexity has become a vital concept to describe the chaotic behaviors of a quantum system [1, 2] or a spacetime [3, 4]. The concept of complexity was originally proposed to study the difficulty of a quantum system which can transform from one state to another. Therefore, it provides a new insight to probe the chaotic behaviors of a quantum system. In the spacetime, it was suggested that the complexity of a quantum state on the boundary is dual to the volume of the Einstein-Rosen (ER) bridge, which is the so-called CV conjecture [3, 4]. This kind of duality is one of the manifestations of holography [5].

As a matter of fact, there are various kinds of complexity. For instance, in the quantum computings, Nielson et.al. have introduced the notion of quantum ”circuit complexity” to evaluate the minimum numbers of logical steps which is required to perform a task [1, 6, 7]. Interestingly, this kind of quantum circuit complexity is closed related to the minimal length of paths in a curved geometry, which is a sign of the correspondence between the complexity and certain quantities in geometry. Another method for calculating complexity of states in quantum field theory is based on the “Fubini-Study” metric from the information geometry [8]. It should be noted that both of the above approaches depend on the choice of the parametric manifolds.

Compared to the above two approaches, the Krylov complexity is free of ambiguities without needing the parametric manifolds in light of its definition [2]. Krylov complexity was established to explore the growth of operators in quantum systems and to distinguish between the chaotic and integrable systems. Due to its uniqueness, Ref. [2] proposed a universal bound for the Krylov complexity according to the specific Lanczos coefficients. Later on, the concept of Krylov complexity has been extended into a large number of fields, for instance in: the Sachdev-Ye-Kitaev (SYK) model [9, 10], the generalized coherent state [11], the Ising and Heisenberg models [12, 13, 14], conformal field theories [15, 16], topological phases of matter [17], integrable models with saddle-point dominated scrambling [18], cosmological complexity [19], thermal quantum field theories [20, 21, 22] and $\it e.t.c$ . The very recent developments can be found in Refs. [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. And there is a very comprehensive review of Krylov complexity [45].

There have been several efforts to incorporate Krylov complexity into the framework of holography. For instance, Ref. [46] established a one-to-one correspondence between the thermo-field double (TFD) state (at infinite temperature) and Jackiw-Teitelboim (JT) gravity [47, 48]. Then, the Krylov complexity can be computed in the SYK model using JT gravity through holography. Additionally, Ref. [49] evaluated the evolution of Krylov complexity in JT gravity using the random matrix theory. It is well-known that the information geometry connects geometry to quantum states through the Fisher metric (or Fubini-Study metric) [50]. Therefore, from this perspective, Ref. [51] systematically investigated the geometry of Krylov complexity and concluded that $V_{t}=2\pi K_{\mathcal{O}}$ where $V_{t}$ is the volume of the Fubini-Study metric and $K_{\mathcal{O}}$ is the Krylov complexity of the operator $\mathcal{O}$ . However, they only investigated the closed system with single mode.

In this letter, we propose the generalized CV conjecture in the closed and open systems from the Hermitian Hamiltonian. In order to test our conjecture, we study the two-mode Hermitian Hamiltonian. For the closed system, the wave function is constructed from the two-mode squeezed state by employing the displacement operators. As for the open system, the construction of the wave functions is more involved, which needs the second kind of Meixner polynomials [52, 2]. The resulting wave function is dubbed open two-mode squeezed state. Eventually, we observe that the generalized CV conjectures always hold in both closed and open systems, which provide strong evidences for the equality $V_{t}=2\pi K_{\mathcal{O}}$ to the Hermitian Hamiltonian.

II Lanczos Algorithm

The Lanczos algorithm is based on a specific recurrence relation [53], particularly satisfied with the second kind of Meixner polynomials [54]. The implementation of the Lanczos algorithm depends on the quantum state represented by $\mathcal{O}_{n}$ . In our context, it is more convenient to operate within the Heisenberg picture, characterized by the equation $\partial_{t}\mathcal{O}(t)=i[H,\mathcal{O}(t)]$ . During the process, we utilize the Liouvillian superoperator $\mathcal{L}_{X}$ , defined as $\mathcal{L}_{X}Y=[X,Y]$ , where $[\ ,\ ]$ denotes the commutator. As discussed in [55], the Liouvillian superoperator $\mathcal{L}_{X}$ can be interpreted as the Hamiltonian in terms of creation and annihilation operators. The Lanczos algorithm constructs the orthogonal basis $\{\mathcal{O}_{n}\}$ due to the Gram-Schmidt method satisfying with

\mathcal{L}|\mathcal{O}_{n})=b_{n+1}|\mathcal{O}_{n+1})+b_{n}|\mathcal{O}_{n-1}),

(1)

where $\mathcal{L}\equiv\mathcal{L}_{X}$ for simplcity and $b_{n}$ denotes the Lanczos coefficient, which plays a crucial role in determining the nature of the dynamical system, such as whether it is chaotic, integrable, or free. For further details on the Lanczos coefficient, please refer to Ref. [55]. In this study, we focus on the generalized Lanczos algorithm [56], which is derived from the Lindblad master equation [57]. The generalized recurrence relation associated with this algorithm can be expressed as follows:

\mathcal{L}|\mathcal{O}_{n})=-ic_{n}|\mathcal{O}_{n})+b_{n+1}|\mathcal{O}_{n+1})+b_{n}|\mathcal{O}_{n-1}),

(2)

where we will follow the notation of our previous work [52, 58]. We followed a specific notation from the perspective of the matrix representation of quantum mechanics. Here, $ic_{n}$ arises from the diagonal component of the matrix. Based on the work of [56], we observed that the $\mathcal{O}_{n}$ part corresponds to the open system, while $\mathcal{O}_{n+1}$ and $\mathcal{O}_{n-1}$ relate to the closed system. From this perspective, the generalized Lanczos algorithm can extract information from a specific Hamiltonian encoded in $ic_{n}$ , similar to our earlier studies [59, 60, 58, 52]. More specifically, once the Hamiltonian is established, the generalized Lanczos algorithm (2) can be employed to distinguish which components represent an open system and which parts represent a closed system. Reference [58] even discusses the possibility that a Hermitian Hamiltonian could exhibit characteristics of an open system.

In this letter, we will focus on the most general two-mode Hamiltonian concerning the creation and annihilation operators, ensuring Hamiltonian is Hermitian and satisfies with the generalized Lanczos algorithm, where the two-mode quantum state can be represented by $|\mathcal{O}_{n})=|n;n\rangle_{\vec{k},-\vec{k}}$ , where $n$ denotes the $n$ -th excited state and $\vec{k}$ is the momentum.

III Wave function

The essence of the generalized Lanczos algorithm (2) is a recurrence relation belonging to the second kind of Meixner polynomials [53, 54]. Equation (2) can be rewritten in the form of an orthogonal polynomial sequence (OPS):

P_{n+1}(x)=(x-\tilde{c}_{n})P_{n}(x)-b_{n}^{2}P_{n-1}(x),

(3)

where we define $\tilde{c}_{n}=-ic_{n}$ , and $x$ represents the Hamiltonian. The sequence begins with the initial conditions $P_{0}(x)=1$ and $P_{1}(x)=x-c_{0}$ . The polynomial $P_{n}(x)=\det(x-\mathcal{L}_{n})$ is for an open system, and $\mathcal{L}_{n}$ is the Liouvillian superoperator for the $n$ -th quantum state. By introducing the natural orthonormal basis $\{e_{n}\}$ . We could represent $|P_{n}(x))=\Bigl{(}\prod_{k=1}^{n}b_{k}\Bigr{)}|\mathcal{O}_{n}),\ \ \mbox{and}\ \ \ |x^{n})=\mathcal{L}^{n}|\mathcal{O}).$ Combining Eq. (3), there is an explicit relation as follows,

b_{n+1}e_{n+1}+b_{n}e_{n-1}=(x-\tilde{c}_{n})e_{n},

(4)

where $e_{n}=\mathcal{O}_{n}$ in our case and it is equivalent to the generalized Lanczos algorithm (2). Being armed with generating function of Meixner polynomials, the corresponding wave function can be derived by (the details can be found in Ref. [52])

			$\displaystyle\|\mathcal{O}(\eta))=e^{i\mathcal{L}\eta}\|e_{0})$
	$\displaystyle=$		$\displaystyle\frac{{\rm sech}~{}r_{k}}{1+u_{2}\tanh r_{k}}\sum_{n=0}^{\infty}\|1-u_{1}^{2}\|^{\frac{n}{2}}\frac{(-\exp(2i\phi_{k}\tanh r_{k})^{n}}{(1+u_{2}\tanh r_{k})^{n}}\|e_{n}).$

where $r_{k}\equiv r_{k}(\eta)$ , $\phi_{k}\equiv\phi_{k}(\eta)$ are the parameters depicting the feature of two-mode squeezed state. The parameters $u_{1}$ and $u_{2}$ are determined by Hamiltonian. This wave function is dubbed as the open two-mode squeezed state since the leading order in terms of $u_{2}$ is exactly the two-mode squeezed state

			$\displaystyle\|\mathcal{O}(\eta))=\hat{S}_{\vec{k}}(r_{k},\phi_{k})\|0;0\rangle_{\vec{k},-\vec{k}}$
	$\displaystyle=$		$\displaystyle\frac{1}{\cosh r_{k}}\sum_{n=0}^{\infty}(-1)^{n}e^{2in\phi_{k}}\tanh^{n}r_{k}\|n;n\rangle_{\vec{k},-\vec{k}},$		(6)

$\hat{S}_{\vec{k}}$ plays a role of displacement operator.

The derivation of this wave function is highly significant as it is model-independent, relying only on the recurrence relation (3). Information of various models is encoded in the parameters $u_{1}$ and $u_{2}$ , with $u_{2}$ serving as a dissipative coefficient, as noted in Refs. [58, 60, 52]. Although our wave function is derived within the framework of cosmological perturbation theory for the inflation, it can also be applied to fields such as condensed matter, quantum information, quantum optics, and $\it etc$ [45].

IV The generalized CV conjecture

The CV conjecture arises from the analysis of the complexity of quantum states and is related to the ER bridge within the holographic framework [4]. It is reasonable to extend this conjecture to broader quantum systems. In [51], the geometry of Krylov complexity is examined using the Fubini-Study metric. The authors concluded that $V_{g}=2\pi\mathcal{C}_{K}$ , where $V_{g}$ represents the volume of the Fubini-Study metric and $\mathcal{C}_{K}$ denotes the Krylov complexity. However, their findings are applicable only to closed systems, since the Liouvillian superoperator only contributes to the Lanczos coefficient part ( $b_{n}$ part) with single mode. Nevertheless, Ref. [51] provides a strong hint that we could extend the CV conjecture within the framework of information geometry, as it explicitly connects probability to Riemannian geometry.

In this letter, we assume that $V_{g}=2\pi\mathcal{C}_{K}$ remains valid for the general Hermitian Hamiltonian, which includes the open system part proportional to $c_{n}$ . The property of Hermitian will guide us to construct the general two-mode Hamiltonian.

IV.1 Generalized CV conjecture for closed system

To test this generalized CV conjecture, we will first focus on the part of closed system. In particular, we will examine the two-mode Hermitian Hamiltonian, whose group representation is the Weyl algebra. Let us recall that the methology of [51], where the key ingredient is the generalized displacement operator according to the Hamiltonian. For various groups, its representation theory is different. The creation and annihilation operators of every group are also distinctive, leading the different wave functions.

In light of this logic, we see that the excited state of the wave function for the Weyl algebra correspond to the well-known two-mode squeezed state. Additionally, we can recognize that the squeezed operator serves as the generalized displacement operator for the two-mode Weyl algebra. We will continue applying this method for the two-mode system. First, we can write down the displacement operator as

D(\eta)=S(\eta)=e^{iH\eta}=\exp(-\eta\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}^{\dagger}+\bar{\eta}\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}),

(7)

where $\eta$ is a complex number, $\bar{\eta}$ is the complex conjugate of $\eta$ . Then, using the $\hat{a}^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle,\ \hat{a}|n\rangle=\sqrt{n}|n-1\rangle$ , and changing the variable $\eta=re^{2i\phi}$ , we can explicitly obtain the two-mode squeezed state (6). According to the definition of Krylov complexity $K_{\mathcal{O}}=\sum_{n}n|\varphi_{n}(t)|^{2}$ , we could obtain the Krylov complexity as,

K_{\mathcal{O}}=\sinh^{2}r_{k}.

(8)

To relate CV conjecture, we need Fubini-Study metric whose definition is,

ds_{FS}^{2}=\frac{\langle dz|dz\rangle}{\langle z|z\rangle}-\frac{\langle dz|z\rangle\langle z|dz\rangle}{\langle z|z\rangle^{2}}.

(9)

where $|z\rangle=|\mathcal{O}_{n})$ . Under the complex coordinates $(r,\phi)$ , the metric of two-mode squeezed state can be explicitly obtained as follows,

ds^{2}=\frac{dzd\bar{z}}{(1-z\bar{z})^{2}}=dr^{2}+\sinh^{2}2rd\phi^{2}.

(10)

Consequently, one can easily obtain the corresponding volume of this metric,

V_{t}=\int_{0}^{r_{k}}dr\int_{0}^{2\pi}d\phi\sqrt{g}=2\pi\sinh^{2}r_{k}=2\pi K_{\mathcal{O}},

(11)

where $r_{k}$ should be positive according to the physical meaning of two-mode squeezed state. Thus, we could see that the conjecture $V_{t}=2\pi K_{\mathcal{O}}$ is valid for the closed system within two-mode Hermitian Hamiltonian.

IV.2 Generalized CV conjecture in open system

The Hermitian Hamiltonian of an open system can take various forms. Ref. [61] has systematically demonstrated that two-mode squeezed states belong to the “two-photon” algebra, which is spanned by specific operators

\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger},\ \ \ \hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{1}{2}\delta_{ij},\ \ \ \hat{a}_{i}\hat{a}_{j},\ \ \ \hat{a}_{i}^{\dagger},\ \ \ \hat{a}_{i},\ \ \ I,

(12)

where $i,j=\vec{k},-\vec{k}$ represent the two modes. With these operators defined, we need to be more specific in constructing the Hamiltonian. Initially, we focus on the portion of the open system that generates $c_{n}|\mathcal{O}_{n})$ , characterized by $\hat{a}^{\dagger}_{i}\hat{a}_{j}+\frac{1}{2}\delta_{ij}$ . It’s important to note that $\hat{a}^{\dagger}_{i}\hat{a}_{j}$ should act on the same mode, which is referred to as the number operator $N_{i}=\hat{a}^{\dagger}_{i}\hat{a}_{i}$ . Regarding the identity matrix, it can be simplified using the commutation relation $[a_{i},a_{j}^{\dagger}]=\delta_{ij}$ . For the terms $b_{n+1}|\mathcal{O}_{n+1})+b_{n}|\mathcal{O}_{n-1})$ , we can derive that $\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}$ and $\hat{a}_{i}\hat{a}_{j}$ should act on the two different modes. Based on these considerations, the general Hermitian Hamiltonian with two modes can be expressed as:

H=u_{2}(\hat{a}_{\vec{k}}^{\dagger}\vec{a}_{\vec{k}}+\vec{a}_{-\vec{k}}\hat{a}_{-{\vec{k}}}^{\dagger})+\xi\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}^{\dagger}+\bar{\xi}\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}.

(13)

where $u_{2}$ represents the dissipative coefficient. This Hamiltonian indicates that it is impossible to define the generalized displacement operator due to the presence of the term $u_{2}(\hat{a}_{\vec{k}}^{\dagger}\vec{a}_{\vec{k}}+\vec{a}_{-\vec{k}}\hat{a}_{-{\vec{k}}}^{\dagger})$ . To construct the corresponding wave function, we need to utilize the second kind of Meixner polynomials, as discussed in [52]. The resulting wave function is given by (LABEL:open_two_mode_state).

To test $V_{t}=2\pi K_{\mathcal{O}}$ , we need to calculate the Fubini-Study metric. In light of the open two-mode squeezed state (LABEL:open_two_mode_state), we could exlicitly obtain the Krylov complexity as,

K_{\mathcal{O}}=\frac{|1-u_{1}^{2}|\tanh^{2}t}{1+2u_{2}\tanh t+(u_{2}^{2}-|1-u_{1}^{2}|)\tanh^{2}t}.

(14)

where we have normalized the wave function for the Krylov complexity $K_{\mathcal{O}}=\frac{1}{(\mathcal{O}(t)|\mathcal{O}(t))}\sum_{n}n|\varphi_{n}(t)|^{2}$ . The formula (14) is consistent with [56].

Next, we will use the Fubini-Study metric to evaluate the generalized CV conjecture. In our calculations, we will consider $u_{1}$ and $u_{2}$ as time-independent variables, determined by various models. For convenience, we define $|z\rangle=|\mathcal{O}(\eta))$ . After some algebra, we can derive the corresponding Fubini-Study metric as follows (details can be found in Appendix A):

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\Bigl{(}\frac{2}{\sinh 2r}-\frac{u_{2}\rm sech^{2}r}{1+u_{2}\tanh r}\Bigr{)}^{2}\frac{A}{(1-A)^{2}}dr^{2}$		(15)
			$\displaystyle+4\frac{A}{(1-A)^{2}}d\phi^{2},$		(15)

with $A=|1-u_{1}^{2}|\tanh^{2}r/(1+u_{2}\tanh r)^{2}$ and we can obtain $\frac{A}{(1-A)^{2}}=\frac{|1-u_{1}^{2}|\tanh^{2}r(1+u_{2}\tanh r)^{2}}{[1+2u_{2}\tanh r+(u_{2}^{2}-|1-u_{1}^{2}|)\tanh^{2}r]^{2}}$ . With these formulae, we can obtain the square root of the metric determinant as,

$\displaystyle\sqrt{g}$	$\displaystyle=$	$\displaystyle\Bigl{\|}\frac{2}{\sinh 2r}-\frac{u_{2}\rm sech^{2}r}{1+u_{2}\tanh r}\Bigr{\|}\frac{2A}{(1-A)^{2}}$	(16)
	$\displaystyle=$	$\displaystyle\Bigl{\|}\frac{2}{\sinh 2r}-\frac{u_{2}\rm sech^{2}r}{1+u_{2}\tanh r}\Bigr{\|}\times$
		$\displaystyle\frac{2\|1-u_{1}^{2}\|\tanh^{2}r(1+u_{2}\tanh r)^{2}}{[1+2u_{2}\tanh r+(u_{2}^{2}-\|1-u_{1}^{2}\|)\tanh^{2}r]^{2}}.$

Remarkably, we find that the generalized conjecture is still hold for the open system with Hermitian conditions,

	$\displaystyle V_{t}$	$\displaystyle=$	$\displaystyle\int_{0}^{t}dr\int_{0}^{2\pi}d\phi\sqrt{g}$
		$\displaystyle=$	$\displaystyle 2\pi\frac{\|1-u_{1}^{2}\|\tanh^{2}t}{1+2u_{2}\tanh t+(u_{2}^{2}-\|1-u_{1}^{2}\|)\tanh^{2}t}=2\pi K_{\mathcal{O}}.$

Until now, we have tested the generalized CV conjecture for the most general two-mode Hamiltonian, which involves creation and annihilation operators. Additionally, we can conjecture that the generalized CV conjecture is valid for Hermitian Hamiltonians, which encompass both closed and open system components.

V Conclusions and outlook

The work by [51] has constructed wave functions using the generalized displacement operator, where the representation theory of distinct groups generates various displacement operators. Moreover, they discovered that the Krylov complexity of different quantum states is proportional to the volume of the corresponding Fubini-Study metric, although their analysis is limited to closed systems with a single mode. In light of this logic, we propose the generalized CV conjecture to quantum states generated by displacement operators within the framework of information geometry. Specifically, we conjecture that $V_{t}=2\pi K_{\mathcal{O}}$ , which is applicable to Hermitian Hamiltonians, encompassing both closed and open system components. To test our conjecture, we examined the general two-mode Hermitian Hamiltonian, characterized by the Weyl algebra. For the closed part, the wave function is generated using the generalized displacement operator, as demonstrated in Eq. (7). Here, we can explicitly prove that $V_{t}=2\pi K_{\mathcal{O}}$ , as shown in (11). However, when considering the open system component, the wave function cannot be derived using the approach from [51]. Instead, we utilize the second kind of Meixner polynomials for construction, as indicated in Eq. (LABEL:open_two_mode_state). After some complex computations, we remarkably find that $V_{t}=2\pi K_{\mathcal{O}}$ still holds true.

The reader might note that our wave function, as presented in equation (LABEL:open_two_mode_state), is rather unique. However, we have established that this wave function is applicable to the general two-mode Hermitian Hamiltonian, which can be utilized in various fields such as quantum information, quantum optics, condensed matter, and $\it etc$ . It is important to point out that our generalized CV conjecture is only valid for the two-dimensional Fubini-Study metric. Hopefully, we can extend our conjecture to higher-dimensional cases. Recently, Ref. [46, 62] proposed that the Krylov complexity of double-scaled SYK is explicitly linked to the length of the two-sided wormhole in $AdS_{2}$ . This naturally aligns with our methodology and the wave function related to JT gravity.

VI acknowledgments

LHL and KHZ are funded by National Natural Science Foundation of China (NSFC) with grant NO. 12165009, Hunan Natural Science Foundation with grant NO. 2023JJ30487 and NO. 2022JJ40340. HQZ is funded by NSFC with grant NO. 12175008.

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Appendix A The calculation of the metric $ds^{2}$ within the open system

The two-mode squeezed state within the open system is written as

|z\rangle=\frac{{\rm sech}\ r}{1+u_{2}\tanh r}\sum_{n=0}^{\infty}|1-u_{1}^{2}|^{\frac{n}{2}}\Bigl{(}\frac{-e^{2i\phi}\tanh r}{1+u_{2}\tanh r}\Bigr{)}^{n}|n_{\vec{k}};n_{-\vec{k}}\rangle.

(18)

Next, we calculate the inner product of them as

\langle z|z\rangle=\Bigl{(}\frac{{\rm sech}\ r}{1+u_{2}\tanh r}\Bigr{)}^{2}\frac{1}{1-A},

(19)

where $A=|1-u_{1}^{2}|\tanh^{2}r/(1+u_{2}\tanh r)^{2}$ is utilized in the subsequent calculation for simplification. The following items that we need to be aware of

\langle z|n|z\rangle=\Bigl{(}\frac{{\rm sech}\ r}{1+u_{2}\tanh r}\Bigr{)}^{2}\frac{1}{(1-A)^{2}},

(20)

and another one as

\langle z|n^{2}|z\rangle=\Bigl{(}\frac{{\rm sech}\ r}{1+u_{2}\tanh r}\Bigr{)}^{2}\frac{A+A^{2}}{(1-A)^{3}}.

(21)

Thirdly, the term $|dz\rangle$ is written as

\displaystyle|dz\rangle=2ind\phi|z\rangle+\Bigl{(}-\frac{\sinh 2r+2u_{2}\cosh^{2}r}{2\cosh^{2}r(1+u_{2}\tanh r)}+\frac{2n}{\sinh 2r}-\frac{nu_{2}{\rm sech^{2}}r}{1+u_{2}\tanh r}\Bigr{)}dr|z\rangle.

(22)

Finally, we obtain the information metric within the Hermitian open system as

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\ \Bigl{[}\Bigl{(}\frac{2}{\sinh 2r}-\frac{u_{2}{\rm sech^{2}}r}{1+u_{2}\tanh r}\Bigr{)}^{2}dr^{2}+4d\phi^{2}\Bigr{]}\Bigl{(}\frac{\langle z\|n^{2}\|z\rangle}{\langle z\|z\rangle}-\frac{\langle z\|n\|z\rangle^{2}}{\langle z\|z\rangle^{2}}\Bigr{)}$		(23)
		$\displaystyle=$	$\displaystyle\ \Bigl{[}\Bigl{(}\frac{2}{\sinh 2r}-\frac{u_{2}{\rm sech^{2}}r}{1+u_{2}\tanh r}\Bigr{)}^{2}dr^{2}+4d\phi^{2}\Bigr{]}\frac{2A}{(1-A)^{2}}.$		(23)